convergence
https://plus.maths.org/content/taxonomy/term/324
enWhen things get weird with infinite sums
https://plus.maths.org/content/when-things-get-weird-infinite-sums
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Luciano Rila </div>
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/17_dec_2014_-_1009/series_icon.png?1418810995" /> </div>
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<p>What is 1-1+1-1+1-1+...? How infinite sums challenge our notion of arithmetic.</p>
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<div style="float: left; max-width: 570px;">[maths] Things can get weird when we deal with infinity. Consider the following sum
$$S = 1-1+1-1+1-1+1-1+ ...<p><a href="https://plus.maths.org/content/when-things-get-weird-infinite-sums" target="_blank">read more</a></p>https://plus.maths.org/content/when-things-get-weird-infinite-sums#commentsconvergencedivergenceFP-carouselgeometric seriesharmonic seriesinfinite seriesThu, 18 Dec 2014 10:57:19 +0000mf3446264 at https://plus.maths.org/contentInfinity or -1/12?
https://plus.maths.org/content/infinity-or-just-112
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/17_feb_2014_-_1657/icon.jpg?1392656231" /> </div>
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<p>What do you get when you add up all the natural numbers 1+2+3+4+ ... ? Not -1/12! We explore a strange result that has been making the rounds recently.</p>
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<p>Recently a very strange result has been making the rounds. It says
that when you add up all the natural numbers</p><p><a href="https://plus.maths.org/content/infinity-or-just-112" target="_blank">read more</a></p>https://plus.maths.org/content/infinity-or-just-112#commentsconvergenceinfinite seriesRiemann zeta functionTue, 18 Feb 2014 15:32:25 +0000mf3446043 at https://plus.maths.org/contentHow to add up quickly
https://plus.maths.org/content/how-add-quickly
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Chris Budd </div>
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/14_jun_2012_-_0947/icon.jpg?1339663672" /> </div>
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<p>The number pi can be expressed beautifully in terms of infinite sums. For practical purposes though, these sums are rather disappointing: they converge slowly, so you need to sum a large number of terms to get accurate estimates of pi. Here's a clever way to make them converge faster.</p>
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<div style="float: left;width:400px;">[maths]
One of my favourite mathematical results is the famous formula
$$
\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} + \ldots \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \mbox{(1)}$$
As far as I'm concerned, all of maths is here, and if this formula doesn't blow you
away then you simply have no soul. What the formula does is to connect two quite
different concepts, the geometry linked to the number $\pi$ and the simplicity
of the odd numbers.<p><a href="https://plus.maths.org/content/how-add-quickly" target="_blank">read more</a></p>https://plus.maths.org/content/how-add-quickly#commentsconvergenceinfinite seriesTaylor seriesThu, 14 Mar 2013 09:13:54 +0000mf3445711 at https://plus.maths.org/contentNo limits for Usain
https://plus.maths.org/content/no-limits-usain
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Tony Crilly </div>
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/22_jun_2011_-_1254/icon.jpg?1308743645" /> </div>
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<p>Usain Bolt, the "fastest man on the planet", aims to get his 100 metre world record of 9.58 seconds down to 9.40 seconds. What has mathematics got to say about this quest?</p>
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<p>Usain Bolt, the "fastest man on the planet", aims to get his 100 metre world record of 9.58 seconds down to 9.40 seconds. What has mathematics to say about this quest?</p>
<div class="rightimage" style="width: 300px"><img src="/sites/plus.maths.org/files/articles/2011/usain/bolt.jpg" alt="Usain Bolt" width="300" height="384" />
<p>Usain Bolt celebrates his victory over 100m and new world record at the Beijing Olympics. Image: <a href="http://en.wikipedia.org/wiki/File:Boltbeijing.jpg">Jmex60</a>.</p><p><a href="https://plus.maths.org/content/no-limits-usain" target="_blank">read more</a></p>https://plus.maths.org/content/no-limits-usain#commentsconvergencedivergenceharmonic seriesmathematics in sportolympicssequenceFri, 08 Jul 2011 10:11:44 +0000mf3445511 at https://plus.maths.org/contentA disappearing number
https://plus.maths.org/content/disappearing-number
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Rachel Thomas </div>
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/issue49/features/complicite/icon.jpg?1228089600" /> </div>
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Mathematics takes to the stage with <i>A disappearing number</i>, a work by Complicite, inspired by the mathematical collaboration of Hardy and Ramanujan. <b>Rachel Thomas</b> went to see the play, and explains some of the maths. You can also read her <a href="/issue49/interview">interview</a> with Victoria Gould about how the show was created. </div>
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<div class="pub_date">December 2008</div>
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<p><i>One morning early in 1913, he found, among the letters on his breakfast table, a large untidy envelope decorated with Indian stamps. When he opened it, he found sheets of paper by no means fresh, on which, in a non-English holograph, were line after line of symbols. Hardy glanced at them without enthusiasm.<p><a href="https://plus.maths.org/content/disappearing-number" target="_blank">read more</a></p>https://plus.maths.org/content/disappearing-number#comments49convergencedivergenceinfinite seriesmathematics and artmathematics and theatreRiemann zeta functionMon, 01 Dec 2008 00:00:00 +0000plusadmin2346 at https://plus.maths.org/contentCareer interview: Actor and mathematician
https://plus.maths.org/content/career-interview-actor-and-mathematician
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/issue49/interview/icon.jpg?1228089600" /> </div>
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<div class="pub_date">December 2008</div>
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<p><i>A version of this interview is available as a <a href="/podcasts/PlusCareersPodcastDec08.mp3">podcast</a>.</i></p><p><a href="https://plus.maths.org/content/career-interview-actor-and-mathematician" target="_blank">read more</a></p>https://plus.maths.org/content/career-interview-actor-and-mathematician#comments49Arts & Entertainmentcareer interviewconvergencehardyHealth & Societyinfinite seriesmathematics and artmathematics and theatremathematics educationpartitionsramanujanRiemann zeta functionMon, 01 Dec 2008 00:00:00 +0000plusadmin2438 at https://plus.maths.org/contentAn infinite series of surprises
https://plus.maths.org/content/infinite-series-surprises
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C. J. Sangwin </div>
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<img class="imagefield imagefield-field_abs_img" width="130" height="130" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/issue19/features/infseries/icon.jpg?1007164800" /> </div>
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Infinite series occupy a central and important place in mathematics. <b>C. J. Sangwin</b> shows us how eighteenth-century mathematician Leonhard Euler solved one of the foremost infinite series problems of his day. </div>
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<div class="pub_date">Mar 2002</div>
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<h3>Introduction</h3>
[maths]An infinite sum of the form \setcounter{equation}{0} \begin{equation} a_1 + a_2 + a_3 + \cdots = \sum_{k=1}^\infty a_k, \end{equation} is known as an infinite series. Such series appear in many areas of modern mathematics. Much of this topic was developed during the seventeenth century. Leonhard Euler continued this study and in the process solved many important problems.<p><a href="https://plus.maths.org/content/infinite-series-surprises" target="_blank">read more</a></p>https://plus.maths.org/content/infinite-series-surprises#comments19convergencedivergenceEuler's solution to the Basel problemgeometric seriesharmonic seriesinfinite seriesintegral testpower seriesSat, 01 Dec 2001 00:00:00 +0000plusadmin2202 at https://plus.maths.org/contentMathematical mysteries: Zeno's Paradoxes
https://plus.maths.org/content/mathematical-mysteries-zenos-paradoxes
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Rachel Thomas </div>
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/9_jun_2011_-_1545/tort1.jpg?1307630744" /> </div>
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<p>The paradoxes of the philosopher Zeno, born approximately 490 BC in southern Italy, have puzzled mathematicians, scientists and philosophers for millennia. Although none of his work survives today, over 40 paradoxes are attributed to him which appeared in a book he wrote as a defense of the philosophies of his teacher Parmenides.</p>
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<p>The paradoxes of the philosopher <a href="http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Zeno_of_Elea.html">Zeno</a>, born approximately 490 BC in southern Italy, have puzzled mathematicians, scientists and philosophers for millennia. Although none of his work survives today, over 40 paradoxes are attributed to him which appeared in a book he wrote as a defense of the
philosophies of his teacher Parmenides.<p><a href="https://plus.maths.org/content/mathematical-mysteries-zenos-paradoxes" target="_blank">read more</a></p>https://plus.maths.org/content/mathematical-mysteries-zenos-paradoxes#comments17Achilles ParadoxArrow Paradoxconvergencegeometric serieslimitMathematical mysteriesrelativityworldlineZeno's paradoxesFri, 01 Dec 2000 00:00:00 +0000plusadmin4753 at https://plus.maths.org/contentIn perfect harmony
https://plus.maths.org/content/perfect-harmony
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John Webb </div>
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<img class="imagefield imagefield-field_abs_img" width="130" height="130" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/issue12/features/harmonic/icon.jpg?967762800" /> </div>
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The harmonic series is far less widely known than the arithmetic and geometric series. However, it is linked to a good deal of fascinating mathematics, some challenging Olympiad problems, several surprising applications, and even a famous unsolved problem. <b>John Webb</b> applies some divergent thinking, taking in the weather, traffic flow and card shuffling along the way. </div>
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<div class="pub_date">September 2000</div>
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<h2>Introduction</h2>
<p>Two elementary series are studied in school mathematics:</p>
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<li>arithmetic series, such as <img src="/MI/3e35c3a6a00638f7c180b293c8e21feb/images/img-0001.png" alt="$1+2+3+\dots +n$" style="vertical-align:-2px;
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<p>and</p><p><a href="https://plus.maths.org/content/perfect-harmony" target="_blank">read more</a></p>https://plus.maths.org/content/perfect-harmony#comments12arithmetic seriesconvergencedivergencegeometric seriesharmonic serieslogarithmThu, 31 Aug 2000 23:00:00 +0000plusadmin2172 at https://plus.maths.org/contentChaos in Numberland: The secret life of continued fractions
https://plus.maths.org/content/chaos-numberland-secret-life-continued-fractions
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John D. Barrow </div>
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<img class="imagefield imagefield-field_abs_img" width="130" height="130" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/issue11/features/cfractions/icon.jpg?959814000" /> </div>
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One of the most striking and powerful means of presenting numbers is completely ignored in the mathematics that is taught in schools, and it rarely makes an appearance in university courses. Yet the continued fraction is one of the most revealing representations of many numbers, sometimes containing extraordinary patterns and symmetries. <strong>John D. Barrow</strong> explains. </div>
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June 2000
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<p><a href="https://plus.maths.org/content/chaos-numberland-secret-life-continued-fractions" target="_blank">read more</a></p>https://plus.maths.org/content/chaos-numberland-secret-life-continued-fractions#comments11chaoscontinued fractionconvergencegeargolden ratioKhinchin's constantLevy's constantprobability distributionrational approximationWed, 31 May 2000 23:00:00 +0000plusadmin2165 at https://plus.maths.org/content