infinite series
https://plus.maths.org/content/taxonomy/term/505
enMaths in a minute: Writing infinite sums
https://plus.maths.org/content/maths-minute-writing-infinite-sums
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<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_-_1704/infinity_icon.png?1418835845" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<p>How to write a sum that's infinitely long.</p>
</div>
</div>
</div>
[maths] If I tell you to consider the infinite sum
$$1+2+3+4+5...$$
you will know what I mean. It's easy to continue the pattern, it's 6
next, then 7, and so on. The same goes for the infinite sum
$$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + ... $$
But what if the pattern is less obvious?<p><a href="https://plus.maths.org/content/maths-minute-writing-infinite-sums" target="_blank">read more</a></p>https://plus.maths.org/content/maths-minute-writing-infinite-sums#commentsinfinite seriesThu, 18 Dec 2014 16:27:53 +0000mf3446283 at https://plus.maths.org/contentWhen things get weird with infinite sums
https://plus.maths.org/content/when-things-get-weird-infinite-sums
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Luciano Rila </div>
</div>
</div>
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<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>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<p>What is 1-1+1-1+1-1+...? How infinite sums challenge our notion of arithmetic.</p>
</div>
</div>
</div>
<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
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<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>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<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>
</div>
</div>
</div>
<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
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Chris Budd </div>
</div>
</div>
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<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>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<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>
</div>
</div>
</div>
<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/contentComputer geeks break Pi record
https://plus.maths.org/content/computer-geeks-break-pi-record
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/5%20Aug%202010%20-%2017%3A35/icon.jpg?1281026100" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<p>Two computer geeks claim to have calculated the number pi to 5 trillion digits — on a single desktop and in record time. That's 2.3 trillion digits more than the previous world record held by the Frenchman Fabrice Bellard.</p>
</div>
</div>
</div>
<br clear="all">
<div class="rightimage" style="width:300px"><img src="/sites/plus.maths.org/files/news/2010/pi/pi.jpg" width="300" height="225" /></div>
<p>Two computer geeks claim to have calculated the number pi to 5 trillion places, on a single desktop and in record time. That's 2.3 trillion digits more than the previous world record held by the Frenchman Fabrice Bellard. Japanese system engineer Shigeru Kondo and American student Alexander Yee achieved the result using a program created by Yee and a desktop computer built by Kondo.<p><a href="https://plus.maths.org/content/computer-geeks-break-pi-record" target="_blank">read more</a></p>https://plus.maths.org/content/computer-geeks-break-pi-record#commentsformula for Piinfinite seriesThu, 05 Aug 2010 16:26:15 +0000mf3445275 at https://plus.maths.org/contentA disappearing number
https://plus.maths.org/content/disappearing-number
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Rachel Thomas </div>
</div>
</div>
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<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>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
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>
</div>
</div>
<div class="pub_date">December 2008</div>
<!-- plusimport -->
<div class="rightshoutout">Listen to the accompanying <a href="/podcasts/PlusPodcastFeb09.mp3">podcast</a></div>
<br clear="all" />
<div style="color: 80a096; margin-left: 20px; margin-right: 20px;">
<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
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<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>
</div>
</div>
<div class="pub_date">December 2008</div>
<!-- plusimport --><br>
<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/content"Read Euler, read Euler, he is the master of us all."
https://plus.maths.org/content/os/issue42/features/wilson/index
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Robin Wilson </div>
</div>
</div>
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<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/June%2017%2C%202010/icon-8.jpg?1276766189" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
Leonhard Euler was one of the most prolific mathematicians of all time. This year marks the 300th anniversary of his birth. <b>Robin Wilson</b> starts off a four part series on Euler with a look at his life and work. </div>
</div>
</div>
<div class="pub_date">March 2007</div>
<!-- plusimport -->
<br clear="all" />
<p>Leonhard Euler was the most prolific mathematician of all time. He wrote more than 500 books and papers during his lifetime — about 800 pages per year — with an incredible 400 further publications appearing posthumously. His collected works and correspondence are still not completely published: they already fill over seventy large volumes, comprising tens of thousands of pages.</p><p><a href="https://plus.maths.org/content/os/issue42/features/wilson/index" target="_blank">read more</a></p>https://plus.maths.org/content/os/issue42/features/wilson/index#comments42EulerEuler yeargeometryharmonic serieshistory of mathematicsinfinite seriespartition of a numbertopologyThu, 01 Mar 2007 00:00:00 +0000plusadmin2306 at https://plus.maths.org/contentOuter space: The rule of two
https://plus.maths.org/content/outer-space-series
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
John D. Barrow </div>
</div>
</div>
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<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_-_1543/diagram.jpg?1307630599" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
The maths of infinite series </div>
</div>
</div>
<div class="pub_date">September 2005</div>
<!-- plusimport -->
<br clear="all" />
<!-- #echo var="outerspace" -->
<p>Infinities are tricky things and have perplexed mathematicians and philosophers for thousands of years. Sometimes a never-ending list of numbers will become infinitely large; sometimes it will get closer and closer to a definite number; sometimes it will defy having any type of definite limit at all. A little while ago I was giving a talk about "Infinity" that included a look at the simple
geometric series</p><p><a href="https://plus.maths.org/content/outer-space-series" target="_blank">read more</a></p>https://plus.maths.org/content/outer-space-series#comments36infinite seriesinfinityouterspaceWed, 31 Aug 2005 23:00:00 +0000plusadmin4785 at https://plus.maths.org/contentAn infinite series of surprises
https://plus.maths.org/content/infinite-series-surprises
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
C. J. Sangwin </div>
</div>
</div>
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<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>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
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>
</div>
</div>
<div class="pub_date">Mar 2002</div>
<!-- plusimport -->
<br clear="all" />
<!-- no longer used -->
<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/content