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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r1237879475">.mw-parser-output .tmbox{margin:4px 0;border-collapse:collapse;border:1px solid #c0c090;background-color:#f8eaba;box-sizing:border-box}.mw-parser-output .tmbox.mbox-small{font-size:88%;line-height:1.25em}.mw-parser-output .tmbox-speedy{border:2px solid #b32424;background-color:#fee7e6}.mw-parser-output .tmbox-delete{border:2px solid #b32424}.mw-parser-output .tmbox-content{border:2px solid #f28500}.mw-parser-output .tmbox-style{border:2px solid #fc3}.mw-parser-output .tmbox-move{border:2px solid #9932cc}.mw-parser-output .tmbox .mbox-text{border:none;padding:0.25em 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class="plainlinks tmbox tmbox-notice" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/Wikipedia:Did_you_know" title="Did you know"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e0/Symbol_question.svg/40px-Symbol_question.svg.png" decoding="async" width="40" height="41" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e0/Symbol_question.svg/60px-Symbol_question.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e0/Symbol_question.svg/80px-Symbol_question.svg.png 2x" data-file-width="180" data-file-height="185"/></a></span></td><td class="mbox-text">A fact from <b>Grandi's series</b> appeared on Wikipedia's <a href="/wiki/Main_Page" title="Main Page">Main Page</a> in the <b><a href="/wiki/Template:Did_you_know" title="Template:Did you know">Did you know</a></b> column on 12 November 2006. The text of the entry was as follows: <ul><li><i>Did you know</i>... that <b><a href="/wiki/Grandi%27s_series" title="Grandi's series">Grandi's series</a></b> 1 − 1 + 1 − 1 + · · · is <a href="/wiki/Divergent_series" title="Divergent series">divergent</a> and appears to equal <a href="/wiki/0_(number)" class="mw-redirect" title="0 (number)">0</a>, yet in some sense "sums" to <a href="/wiki/One_half" title="One half"><sup>1</sup>⁄<sub>2</sub></a>, producing a paradox once linked to the <a href="/w/index.php?title=Creation_(theology)&action=edit&redlink=1" class="new" title="Creation (theology) (page does not exist)">creation</a> <i><a href="/wiki/Ex_nihilo" class="mw-redirect" title="Ex nihilo">ex nihilo</a></i> of the universe?</li></ul> A record of the entry may be seen at <a href="/wiki/Wikipedia:Recent_additions/2006/November#12_November_2006" title="Wikipedia:Recent additions/2006/November">Wikipedia:Recent additions/2006/November</a>.</td><td class="mbox-imageright"><figure 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If you would like to participate, please visit the project page, where you can join <a href="/wiki/Wikipedia_talk:WikiProject_Mathematics" title="Wikipedia talk:WikiProject Mathematics">the discussion</a> and see a list of open tasks.<span class="metadata wpb-metadata"><span class="wpb-project">Mathematics</span><span class="wpb-project_link">Wikipedia:WikiProject Mathematics</span><span class="wpb-banner_name">Template:WikiProject Mathematics</span><span class="wpb-assessment_cat">mathematics articles</span></span></td><td class="mbox-empty-cell"></td></tr><tr><td class="assess import-low"><a href="/wiki/Category:Low-priority_mathematics_articles" title="Category:Low-priority mathematics articles">Low</a></td><td class="mbox-text" colspan="2">This article has been rated as <b>Low-priority</b> on the <a href="/wiki/Wikipedia:WikiProject_Mathematics/Wikipedia_1.0/Assessment#Priority_scale" class="mw-redirect" title="Wikipedia:WikiProject Mathematics/Wikipedia 1.0/Assessment">project's priority scale</a>.</td></tr></tbody></table></td></tr></tbody></table></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237879475"><table class="plainlinks tmbox tmbox-notice" role="presentation"><tbody><tr><td class="mbox-image"><figure class="mw-halign-none" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/8/80/Camera-photo.svg/40px-Camera-photo.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/80/Camera-photo.svg/60px-Camera-photo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/80/Camera-photo.svg/80px-Camera-photo.svg.png 2x" data-file-width="48" data-file-height="48"/></span><figcaption></figcaption></figure></td><td class="mbox-text">It is requested that an <b>image</b> or <b>photograph</b> of <b>Grandi's series</b> be <a href="/wiki/Wikipedia:Uploading_images" title="Wikipedia:Uploading images">included</a> in this article to <a href="/wiki/Wikipedia:Article_development" title="Wikipedia:Article development">improve its quality</a>. Please replace this template with a more specific <a href="/wiki/Template:Media_request_templates" title="Template:Media request templates">media request template</a> where possible.<br/><small>The <a class="external text" href="https://fist.toolforge.org/fist.php?doit=1&language=en&project=wikipedia&data=Grandi%26%2339%3Bs+series&datatype=articles&params%5Bcatdepth%5D=0&params%5Brandom%5D=50&params%5Bll_max%5D=5&params%5Bcommons_max%5D=5&params%5Bflickr_max%5D=5&params%5Binclude_flickr_id%5D=1&params%5Bwts_max%5D=5&params%5Bgimp_max%5D=5&params%5Besp_max%5D=5&params%5Besp_skip_flickr%5D=1&params%5Bgeograph_max%5D=5&params%5Bforarticles%5D=noimage&params%5Blessthan_images%5D=3&params%5Bdefault_thumbnail_size%5D=&params%5Bjpeg%5D=1&params%5Bpng%5D=1&params%5Bgif%5D=1&params%5Bsvg%5D=1&params%5Bogg%5D=1&params%5Bmin_width%5D=80&params%5Bmin_height%5D=80&sources%5Blanguagelinks%5D=1&sources%5Bcommons%5D=1&sources%5Bflickr%5D=1">Free Image Search Tool</a> or <a rel="nofollow" class="external text" href="https://openverse.org/?q=Grandi%26%2339%3Bs+series&license=cc0,pdm,by,by-sa">Openverse Creative Commons Search</a> may be able to locate suitable images on <a href="/wiki/Flickr" title="Flickr">Flickr</a> and other web sites.</small></td><td class="mbox-imageright"><span class="plainlinks clickbutton"><a class="external text" href="https://commons.wikimedia.org/wiki/special:uploadWizard?&descriptionlang=en&description=%5B%5B%3Aen%3AGrandi%26%2339%3Bs+series%7CGrandi%26%2339%3Bs+series%5D%5D&categories=mathematics"><span class="mw-ui-button mw-ui-progressive">Upload</span></a></span></td></tr></tbody></table> <meta property="mw:PageProp/toc"/> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Talk" data-mw-thread-id="h-Talk-2006-11-10T21:26:00.000Z"><span data-mw-comment-start="" id="h-Talk-2006-11-10T21:26:00.000Z"></span>Talk<span data-mw-comment-end="h-Talk-2006-11-10T21:26:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Grandi%27s_series&action=edit&section=1" title="Edit section: Talk"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Melchoir-2006-11-10T21:26:00.000Z-Talk"></span>Should the section on the Cessaro sum say Holder sum instead? I know they're equivalent, but... <small>—The preceding <a href="/wiki/Wikipedia:Sign_your_posts_on_talk_pages" class="mw-redirect" title="Wikipedia:Sign your posts on talk pages">unsigned</a> comment was added by <a href="/wiki/User:Dchudz" title="User:Dchudz">Dchudz</a> (<a href="/wiki/User_talk:Dchudz" title="User talk:Dchudz">talk</a> • <a href="/wiki/Special:Contributions/Dchudz" title="Special:Contributions/Dchudz">contribs</a>) 19:00, 10 November 2006 (UTC)</small> </p> <dl><dd>Well, my sources call it Cesaro. I'm not familiar with Holder summation, but if it's equivalent then I wouldn't worry about it. <a href="/wiki/User:Melchoir" title="User:Melchoir">Melchoir</a> <a href="https://en.wikipedia.org/wiki/Talk:Grandi%27s_series#c-Melchoir-2006-11-10T21:26:00.000Z-Talk" class="ext-discussiontools-init-timestamplink">21:26, 10 November 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Melchoir-2006-11-10T21:26:00.000Z-Talk"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Melchoir-2006-11-10T21:26:00.000Z-Talk"></span></dd></dl> <p><span data-mw-comment-start="" id="c-Mstroeck-2006-11-18T18:53:00.000Z-Talk"></span>Wow, Melchoir, thank you for writing this article! <a href="/wiki/User:Mstroeck" title="User:Mstroeck">mstroeck</a> <a href="https://en.wikipedia.org/wiki/Talk:Grandi%27s_series#c-Mstroeck-2006-11-18T18:53:00.000Z-Talk" class="ext-discussiontools-init-timestamplink">18:53, 18 November 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Mstroeck-2006-11-18T18:53:00.000Z-Talk"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Mstroeck-2006-11-18T18:53:00.000Z-Talk"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Melchoir-2006-11-18T19:13:00.000Z-Mstroeck-2006-11-18T18:53:00.000Z"></span>Ja! <a href="/wiki/User:Melchoir" title="User:Melchoir">Melchoir</a> <a href="https://en.wikipedia.org/wiki/Talk:Grandi%27s_series#c-Melchoir-2006-11-18T19:13:00.000Z-Mstroeck-2006-11-18T18:53:00.000Z" class="ext-discussiontools-init-timestamplink">19:13, 18 November 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Melchoir-2006-11-18T19:13:00.000Z-Mstroeck-2006-11-18T18:53:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Melchoir-2006-11-18T19:13:00.000Z-Mstroeck-2006-11-18T18:53:00.000Z"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Sort_out_references" data-mw-thread-id="h-Sort_out_references-2006-12-08T20:53:00.000Z"><span data-mw-comment-start="" id="h-Sort_out_references-2006-12-08T20:53:00.000Z"></span>Sort out references<span data-mw-comment-end="h-Sort_out_references-2006-12-08T20:53:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Grandi%27s_series&action=edit&section=2" title="Edit section: Sort out references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Melchoir-2006-12-08T20:53:00.000Z-Sort_out_references"></span>For the record, I'm commenting here that I'm now going to drop some references and their claims, and I'm re-integrating the References section. It's become clear that certain facts are repeated by multiple authors and traceable to primary sources, and some claims are just made up, and now that I've started identifying the difference, there is little point in including the latter. Books that don't claim to be reliable historical sources don't need us to emphasize that point. <a href="/wiki/User:Melchoir" title="User:Melchoir">Melchoir</a> <a href="https://en.wikipedia.org/wiki/Talk:Grandi%27s_series#c-Melchoir-2006-12-08T20:53:00.000Z-Sort_out_references" class="ext-discussiontools-init-timestamplink">20:53, 8 December 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Melchoir-2006-12-08T20:53:00.000Z-Sort_out_references"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Melchoir-2006-12-08T20:53:00.000Z-Sort_out_references"></span> </p><p><br/> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Some_information" data-mw-thread-id="h-Some_information-2006-12-09T02:22:00.000Z"><span data-mw-comment-start="" id="h-Some_information-2006-12-09T02:22:00.000Z"></span>Some information<span data-mw-comment-end="h-Some_information-2006-12-09T02:22:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Grandi%27s_series&action=edit&section=3" title="Edit section: Some information"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Melchoir-2006-12-09T02:22:00.000Z-Some_information"></span>Concerning Hölder and Cesàro, for details one is referred to </p> <ul><li>E.W. Hobson, <i>The theory of functions of a real variable and the theory of Fourier's series</i>, Volume II, 2nd edition, reprinted (Cambridge University Press, 1950), section 44.</li></ul> <p>Historically, Hölder's approach dates from 1882 (<i>Math. Annalen, Vol. XX, (1882),</i> p. 535) and Cesàro's from 1890 (<i>Bulletin des Sciences Math., Vol. XIV (1890)</i>, p. 114). It was Knopp (<i>Grenzwerte von Reihen bei der Annäherung an die Convergenzgrenze</i>, Inaugural Dissertation, Berlin 1907) who first showed that a series which is summable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (H,k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>H</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (H,k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10bcc6f942beb8c198781731955349c2e833137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.699ex; height:2.843ex;" alt="{\displaystyle \ (H,k)}"/></span> (that is summable in accordance with Hölder's definition, of order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c682e4c73906031c21d3ec3951990a6111eeacb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle \ k}"/></span>) is necessarily summable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (C,k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>C</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (C,k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c901fa6fd3eaf9402b8e19608ed8a7c6e7a8c9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.401ex; height:2.843ex;" alt="{\displaystyle \ (C,k)}"/></span> (that is summable in accordance with Cesàro's definition, of order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c682e4c73906031c21d3ec3951990a6111eeacb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle \ k}"/></span>). The converse was established by Schnee (<i>Math. Annalen, Vol. LXVII (1909)</i>, p. 110) and by Ford (<i>Amer. Journ. of Math., Vol. XXXII (1910)</i>, p. 315). </p><p>You may take the above information over in your reworking of the present and other related articles. </p><p>--BF 02:01, 9 December 2006 (UTC) </p> <dl><dd>Thanks for the info! In the last few weeks, I've learned some of this, but I didn't know the historical details.</dd> <dd>I'm not sure if this article should talk about the higher-order methods, but they should definitely appear in the more dedicated articles.</dd> <dd>One thing worries me: what exactly did Hölder propose in 1882? One or two of my sources claim that the 1890 Cesàro sum was the first systematic method of summation beyond the convergent one. It's implied that theorems by Abel and Frobenius that we now think of as relating different summation methods weren't posed as such at the time, and early efforts by Leibniz and Euler don't count because they weren't stated in precise terms. So it's conceivable to me that Hölder's work was along the same lines; on the other hand, it could be a scoop of Cesàro. Anyone know? <a href="/wiki/User:Melchoir" title="User:Melchoir">Melchoir</a> <a href="https://en.wikipedia.org/wiki/Talk:Grandi%27s_series#c-Melchoir-2006-12-09T02:22:00.000Z-Some_information" class="ext-discussiontools-init-timestamplink">02:22, 9 December 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Melchoir-2006-12-09T02:22:00.000Z-Some_information"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Melchoir-2006-12-09T02:22:00.000Z-Some_information"></span></dd></dl> <p><br/><span data-mw-comment-start="" id="c-Melchoir-2006-12-09T23:14:00.000Z-Some_information"></span> You are welcome. I would recommend you to consult the book by Hobson cited above (Dover has reprinted this book in 1957). </p><p>Briefly, <b>the Hölder and Cesàro techniques are in principle distinct</b>; it was not until 1909 that one knew that a series is summable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (H,k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>H</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (H,k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10bcc6f942beb8c198781731955349c2e833137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.699ex; height:2.843ex;" alt="{\displaystyle \ (H,k)}"/></span> if and only if it is summable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (C,k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>C</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (C,k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c901fa6fd3eaf9402b8e19608ed8a7c6e7a8c9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.401ex; height:2.843ex;" alt="{\displaystyle \ (C,k)}"/></span> and not until 1913 that one knew that the two methods are <b>equivalent</b>. </p><p>For completeness, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}+a_{2}+\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}+a_{2}+\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76049eff7ab4e80eac39c17f921096546e97df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.972ex; height:2.343ex;" alt="{\displaystyle a_{1}+a_{2}+\dots }"/></span> be the series to be considered. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n}=a_{1}+\dots +a_{n},n=1,2,\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n}=a_{1}+\dots +a_{n},n=1,2,\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c2d53c02337e7c34a05b7fd7019273ea9554f14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.187ex; height:2.509ex;" alt="{\displaystyle s_{n}=a_{1}+\dots +a_{n},n=1,2,\dots }"/></span>. The Hölder partial sums are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{h_{n}^{(k)}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{h_{n}^{(k)}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc2f39286537dd9ba7ba6579f0c2ca9e3304a47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.032ex; height:3.509ex;" alt="{\displaystyle \{h_{n}^{(k)}\}}"/></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{n}^{(1)}=(s_{1}+\dots +s_{n})/n,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{n}^{(1)}=(s_{1}+\dots +s_{n})/n,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e739812899588ac290fa0fc8fbe9369a42d18506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.642ex; height:3.509ex;" alt="{\displaystyle h_{n}^{(1)}=(s_{1}+\dots +s_{n})/n,}"/></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{n}^{(k)}=(h_{1}^{(k-1)}+\dots +h_{n}^{(k-1)})/n,k=2,3,\dots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{n}^{(k)}=(h_{1}^{(k-1)}+\dots +h_{n}^{(k-1)})/n,k=2,3,\dots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe244c0bb235ec79cb59f44118a8ab6693037ce5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.685ex; height:3.676ex;" alt="{\displaystyle h_{n}^{(k)}=(h_{1}^{(k-1)}+\dots +h_{n}^{(k-1)})/n,k=2,3,\dots .}"/></span> The Cesàro partial sums are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{C_{n}^{(k)}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{C_{n}^{(k)}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a77efbe45c836da5d0b3958900533b80159fa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.491ex; height:3.509ex;" alt="{\displaystyle \{C_{n}^{(k)}\}}"/></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n}^{(k)}=[k!(n-1)!/(n+k-1)!]s_{n}^{(k)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">[</mo> <mi>k</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo stretchy="false">]</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n}^{(k)}=[k!(n-1)!/(n+k-1)!]s_{n}^{(k)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42bdf73883299322f2e9d40fd2244d4d8ed04058" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.796ex; height:3.509ex;" alt="{\displaystyle C_{n}^{(k)}=[k!(n-1)!/(n+k-1)!]s_{n}^{(k)}}"/></span> in which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n}^{(0)}=s_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n}^{(0)}=s_{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdc066136e337ce60b61502348e218f06abec320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.478ex; height:3.343ex;" alt="{\displaystyle s_{n}^{(0)}=s_{n},}"/></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n}^{(1)}=s_{1}^{(0)}+\dots +s_{n}^{(0)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n}^{(1)}=s_{1}^{(0)}+\dots +s_{n}^{(0)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730fe4d6975a1f5870e10ebdff5357028c024464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.422ex; height:3.676ex;" alt="{\displaystyle s_{n}^{(1)}=s_{1}^{(0)}+\dots +s_{n}^{(0)},}"/></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n}^{(k)}=s_{1}^{(k-1)}+\dots +s_{n}^{(k-1)},k=2,3,\dots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n}^{(k)}=s_{1}^{(k-1)}+\dots +s_{n}^{(k-1)},k=2,3,\dots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee8084481672bcb3a33d9ff64a9dd659abe8501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.573ex; height:3.676ex;" alt="{\displaystyle s_{n}^{(k)}=s_{1}^{(k-1)}+\dots +s_{n}^{(k-1)},k=2,3,\dots .}"/></span> The above series is summable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (H,k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>H</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (H,k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10bcc6f942beb8c198781731955349c2e833137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.699ex; height:2.843ex;" alt="{\displaystyle \ (H,k)}"/></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }h_{n}^{(k)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }h_{n}^{(k)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c10295585bfdebf394f02fc840ccaae4f5e2d9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.367ex; height:4.509ex;" alt="{\displaystyle \lim _{n\to \infty }h_{n}^{(k)}}"/></span> exists. It is summable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (C,k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>C</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (C,k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c901fa6fd3eaf9402b8e19608ed8a7c6e7a8c9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.401ex; height:2.843ex;" alt="{\displaystyle \ (C,k)}"/></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }C_{n}^{(k)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }C_{n}^{(k)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f31aa85139891cf7c89dd22ac313c15dee258791" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.825ex; height:4.509ex;" alt="{\displaystyle \lim _{n\to \infty }C_{n}^{(k)}}"/></span> exists. It can be shown that when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\lim _{n\to \infty }s_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\lim _{n\to \infty }s_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6f96f0cfc3596fca8b37c7ada51b69d1ddc1a74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.157ex; height:3.676ex;" alt="{\displaystyle s=\lim _{n\to \infty }s_{n}}"/></span> exists, both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }h_{n}^{(k)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }h_{n}^{(k)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c10295585bfdebf394f02fc840ccaae4f5e2d9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.367ex; height:4.509ex;" alt="{\displaystyle \lim _{n\to \infty }h_{n}^{(k)}}"/></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }C_{n}^{(k)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }C_{n}^{(k)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f31aa85139891cf7c89dd22ac313c15dee258791" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.825ex; height:4.509ex;" alt="{\displaystyle \lim _{n\to \infty }C_{n}^{(k)}}"/></span> indeed exist for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c682e4c73906031c21d3ec3951990a6111eeacb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle \ k}"/></span> and are equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/949f49e8db56fbce881a915af8a9f4f9b0ecce2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.671ex; height:1.676ex;" alt="{\displaystyle \ s}"/></span>. Hobson refers to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }s_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }s_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce6bf3104596f21c3bc50d0f325ade1bb4b7fdea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.969ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }s_{n}}"/></span> as <b>ordinary</b> sum and to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }h_{n}^{(k)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }h_{n}^{(k)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c10295585bfdebf394f02fc840ccaae4f5e2d9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.367ex; height:4.509ex;" alt="{\displaystyle \lim _{n\to \infty }h_{n}^{(k)}}"/></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }C_{n}^{(k)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }C_{n}^{(k)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f31aa85139891cf7c89dd22ac313c15dee258791" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.825ex; height:4.509ex;" alt="{\displaystyle \lim _{n\to \infty }C_{n}^{(k)}}"/></span> as <b>conventional</b> sums, which satisfy the <i>consistency condition</i>, that they yield the same result as the ordinary sum when applied to convergent series. </p><p>The proof for the <b>complete equivalence</b> of the two methods of summation (or of summability of order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c682e4c73906031c21d3ec3951990a6111eeacb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle \ k}"/></span>) is given by Schur (<i>Math. Annalen, Vol. LXXIV (1913)</i>, p. 447) and by Hahn (<i>Monatshefte f. Math. u. Physik, Vol. XXXIII (1923)</i>, p. 135). For details (including the proof of the equivalence of the two methods) see sections 55-57 of the above-cited book by Hobson. </p><p>I must admit that there is some confusion in textbooks concerning the two methods. For instance, Whittaker and Watson (4th edition, reprinted, Cambridge University Press, 1962, Sec. 8.43) describe the Cesàro method in exactly the same way as Hobson describes the Hölder method; there is no reference to Hölder's method in the book by Whittaker and Watson. I believe that on this matter Hobson should be considered as the authority. </p><p>--BF 15:13, 9 December 2006 (UTC) </p> <dl><dd>Huh. Actually, yes, I was under the apparently mistaken impression that the Cesàro methods were the iterated ones. By pure luck, it doesn't matter for this article, but if there's confusion to be cleared up then we should do it somewhere. You seem to have good references in front of you; could you be persuaded to start up <a href="/wiki/H%C3%B6lder_summation" title="Hölder summation">Hölder summation</a> and/or expand <a href="/wiki/Ces%C3%A0ro_summation" title="Cesàro summation">Cesàro summation</a> with information on history and higher-order definitions? <a href="/wiki/User:Melchoir" title="User:Melchoir">Melchoir</a> <a href="https://en.wikipedia.org/wiki/Talk:Grandi%27s_series#c-Melchoir-2006-12-09T23:14:00.000Z-Some_information" class="ext-discussiontools-init-timestamplink">23:14, 9 December 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Melchoir-2006-12-09T23:14:00.000Z-Some_information"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Melchoir-2006-12-09T23:14:00.000Z-Some_information"></span></dd></dl> <p><br/><span data-mw-comment-start="" id="c-Melchoir-2006-12-10T18:24:00.000Z-Some_information"></span> Thank you for your suggestion. Expansion of the article on the <a href="/wiki/Ces%C3%A0ro_summation" title="Cesàro summation">Cesàro summation</a> technique is problematic, since one will have to ascribe what at present is ascribed to Cesàro to its rightful author Hölder. This, I believe, should be in the first place the responsibility of the person or persons who have initiated the article on the subject, not least for the fact that this or these persons should have the opportunity to defend their standpoint as expressed in their article. This is why I initiated the present Talk. </p><p>As regards starting up an article on the <a href="/wiki/H%C3%B6lder_summation" title="Hölder summation">Hölder summation</a> method, it seems that this requires, roughly speaking, only renaming the article <a href="/wiki/Ces%C3%A0ro_summation" title="Cesàro summation">Cesàro summation</a> as <a href="/wiki/H%C3%B6lder_summation" title="Hölder summation">Hölder summation</a>. As the two articles are necessarily closely related, I prefer that the entire task be undertaken by the initiator(s) of <a href="/wiki/Ces%C3%A0ro_summation" title="Cesàro summation">Cesàro summation</a>. </p><p>If you wish to modify the article on <a href="/wiki/Ces%C3%A0ro_summation" title="Cesàro summation">Cesàro summation</a> and further to create a page on <a href="/wiki/H%C3%B6lder_summation" title="Hölder summation">Hölder summation</a>, please feel free to do so; in doing so, please also feel free and take over whatever you feel necessary from my earlier texts on this page. In such case, I shall check the two articles and possibly amend them if necessary. In the event that you decide to work on these two articles, please indicate the confusion that exists in the literature of the subject concerning the two summation methods (even despite the fact that they are mathematically equivalent). </p><p>--BF 16:12, 10 December 2006 (UTC) </p> <dl><dd>Sorry, I don't think I'm going to work on those articles myself. FYI, <a href="/wiki/Ces%C3%A0ro_summation" title="Cesàro summation">Cesàro summation</a> was written largely by <a href="/wiki/User:Merge" title="User:Merge">User:Merge</a>, who doesn't seem to be around much. This particular talk page (<a class="mw-selflink selflink">Talk:Grandi's series</a>) is hard to find and seldom read except by me, so if you want to really raise the issue, I encourage you to leave a message at <a href="/wiki/User_talk:Merge" title="User talk:Merge">User talk:Merge</a>, <a href="/wiki/Talk:Ces%C3%A0ro_summation" title="Talk:Cesàro summation">Talk:Cesàro summation</a>, or <a href="/wiki/Wikipedia_talk:WikiProject_Mathematics" title="Wikipedia talk:WikiProject Mathematics">Wikipedia talk:WikiProject Mathematics</a>. Or you can just wait until I make this a Featured Article, at which point lots of people will visit this talk page, but there's no guarantee when that'll happen! <a href="/wiki/User:Melchoir" title="User:Melchoir">Melchoir</a> <a href="https://en.wikipedia.org/wiki/Talk:Grandi%27s_series#c-Melchoir-2006-12-10T18:24:00.000Z-Some_information" class="ext-discussiontools-init-timestamplink">18:24, 10 December 2006 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Melchoir-2006-12-10T18:24:00.000Z-Some_information"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Melchoir-2006-12-10T18:24:00.000Z-Some_information"></span></dd></dl> <p><br/> Thank you for your recommendations. </p><p>--BF 19:21, 10 December 2006 (UTC) </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Merger?" data-mw-thread-id="h-Merger?"><span id="Merger.3F"></span><span data-mw-comment-start="" id="h-Merger?"></span>Merger?<span data-mw-comment-end="h-Merger?"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Grandi%27s_series&action=edit&section=4" title="Edit section: Merger?"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Is iit really necessary to have 5 different articles about Grandi's series? I'm looking at <a class="external free" href="https://en.wikipedia.org/wiki/Grandi%27s_series">https://en.wikipedia.org/wiki/Grandi%27s_series</a> , <a class="external free" href="https://en.wikipedia.org/wiki/History_of_Grandi%27s_series">https://en.wikipedia.org/wiki/History_of_Grandi%27s_series</a> , <a class="external free" href="https://en.wikipedia.org/wiki/Grandi%27s_series_in_education">https://en.wikipedia.org/wiki/Grandi%27s_series_in_education</a> , <a class="external free" href="https://en.wikipedia.org/wiki/Summation_of_Grandi%27s_series">https://en.wikipedia.org/wiki/Summation_of_Grandi%27s_series</a> , and <a class="external free" href="https://en.wikipedia.org/wiki/Occurrences_of_Grandi%27s_series">https://en.wikipedia.org/wiki/Occurrences_of_Grandi%27s_series</a> . Thoe last three don't even have talk pages. I volunteer to help in the merger, but I need permission first. So, a vote? </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Removal_of_"proof"s" data-mw-thread-id="h-Removal_of_"proof"s-2016-10-30T02:33:00.000Z"><span id="Removal_of_.22proof.22s"></span><span data-mw-comment-start="" id="h-Removal_of_"proof"s-2016-10-30T02:33:00.000Z"></span>Removal of "proof"s<span data-mw-comment-end="h-Removal_of_"proof"s-2016-10-30T02:33:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Grandi%27s_series&action=edit&section=5" title="Edit section: Removal of "proof"s"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Jasper_Deng-2016-10-30T02:33:00.000Z-Removal_of_"proof"s"></span>I removed two "proof"s that are nonsensical: the first one using indefinite integrals is invalid because indefinite integrals are only defined up to addition/subtraction of constants, while the second one using a geometric series basically duplicates, in a bad way, what was already given. </p><p>In general, I think this article needs a lot more love. There are rigorous methods of assigning a value to this series, and I agree with the above suggestion that the article on summation be merged here.--<a href="/wiki/User:Jasper_Deng" title="User:Jasper Deng">Jasper Deng</a> <a href="/wiki/User_talk:Jasper_Deng" title="User talk:Jasper Deng">(talk)</a> <a href="https://en.wikipedia.org/wiki/Talk:Grandi%27s_series#c-Jasper_Deng-2016-10-30T02:33:00.000Z-Removal_of_"proof"s" class="ext-discussiontools-init-timestamplink">02:33, 30 October 2016 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Jasper_Deng-2016-10-30T02:33:00.000Z-Removal_of_"proof"s"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Jasper_Deng-2016-10-30T02:33:00.000Z-Removal_of_"proof"s"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="So_what_is_the_value?" data-mw-thread-id="h-So_what_is_the_value?-2018-08-13T01:39:00.000Z"><span id="So_what_is_the_value.3F"></span><span data-mw-comment-start="" id="h-So_what_is_the_value?-2018-08-13T01:39:00.000Z"></span>So what is the value?<span data-mw-comment-end="h-So_what_is_the_value?-2018-08-13T01:39:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:Grandi%27s_series&action=edit&section=6" title="Edit section: So what is the value?"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-2A00:23C5:C10B:A300:A17E:AA1E:4B0:2F91-2018-08-13T01:39:00.000Z-So_what_is_the_value?"></span>After reading the article, I am still no wiser at the value of this series. So what exactly is its value? <a href="/wiki/Special:Contributions/2A00:23C5:C10B:A300:A17E:AA1E:4B0:2F91" title="Special:Contributions/2A00:23C5:C10B:A300:A17E:AA1E:4B0:2F91">2A00:23C5:C10B:A300:A17E:AA1E:4B0:2F91</a> (<a href="/w/index.php?title=User_talk:2A00:23C5:C10B:A300:A17E:AA1E:4B0:2F91&action=edit&redlink=1" class="new" title="User talk:2A00:23C5:C10B:A300:A17E:AA1E:4B0:2F91 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Grandi%27s_series#c-2A00:23C5:C10B:A300:A17E:AA1E:4B0:2F91-2018-08-13T01:39:00.000Z-So_what_is_the_value?" class="ext-discussiontools-init-timestamplink">01:39, 13 August 2018 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-2A00:23C5:C10B:A300:A17E:AA1E:4B0:2F91-2018-08-13T01:39:00.000Z-So_what_is_the_value?"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-2A00:23C5:C10B:A300:A17E:AA1E:4B0:2F91-2018-08-13T01:39:00.000Z-So_what_is_the_value?"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Double_sharp-2019-05-18T02:56:00.000Z-2A00:23C5:C10B:A300:A17E:AA1E:4B0:2F91-2018-08-13T01:39:00.000Z"></span>As the article <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Grandi%27s_series&oldid=848412274">already said</a> back in July 2018, it lacks a sum in the usual sense, but if you want to generalise summation so that it produces a value for this series, the only value that makes sense is 1/2. <a href="/wiki/User:Double_sharp" title="User:Double sharp">Double sharp</a> (<a href="/wiki/User_talk:Double_sharp" title="User talk:Double sharp">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Grandi%27s_series#c-Double_sharp-2019-05-18T02:56:00.000Z-2A00:23C5:C10B:A300:A17E:AA1E:4B0:2F91-2018-08-13T01:39:00.000Z" class="ext-discussiontools-init-timestamplink">02:56, 18 May 2019 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Double_sharp-2019-05-18T02:56:00.000Z-2A00:23C5:C10B:A300:A17E:AA1E:4B0:2F91-2018-08-13T01:39:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Double_sharp-2019-05-18T02:56:00.000Z-2A00:23C5:C10B:A300:A17E:AA1E:4B0:2F91-2018-08-13T01:39:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-Argentum_Kurodil-2020-04-22T10:39:00.000Z-Double_sharp-2019-05-18T02:56:00.000Z"></span>Would 0.5±0.5 not encompass it better?<a href="/w/index.php?title=User:Argentum_Kurodil&action=edit&redlink=1" class="new" title="User:Argentum Kurodil (page does not exist)">Argentum Kurodil</a> (<a href="/wiki/User_talk:Argentum_Kurodil" title="User talk:Argentum Kurodil">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Grandi%27s_series#c-Argentum_Kurodil-2020-04-22T10:39:00.000Z-Double_sharp-2019-05-18T02:56:00.000Z" class="ext-discussiontools-init-timestamplink">10:39, 22 April 2020 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Argentum_Kurodil-2020-04-22T10:39:00.000Z-Double_sharp-2019-05-18T02:56:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Argentum_Kurodil-2020-04-22T10:39:00.000Z-Double_sharp-2019-05-18T02:56:00.000Z"></span></dd></dl></dd></dl>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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