power series
https://plus.maths.org/content/taxonomy/term/509
enMaths in a minute: The power of powers
https://plus.maths.org/content/maths-minute-power-powers
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<p>The powers of <em>x</em> can work magic.</p>
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[maths]Bored of solving quadratic equations? Can't be bothered with cubics? Then it's time to step into the infinite — and marvel at the fact that many of the functions you'll have come across can be expressed using infinite sums made of powers of $x.$
A great example are the trigonometric functions sine and cosine. It turns out that they can be expressed as follows
$$\cos{(x)} = 1 - \frac{x^2}{2!} + \frac{x^4}{4 !} - \frac{x^6}{6!} + ... $$
$$\sin{(x)} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ... ,$$
where $n! = n \times (n-1) \times (n-2) \times ...<p><a href="https://plus.maths.org/content/maths-minute-power-powers" target="_blank">read more</a></p>https://plus.maths.org/content/maths-minute-power-powers#commentscalculuspower seriesTaylor seriesFri, 16 Oct 2015 15:32:55 +0000mf3446434 at https://plus.maths.org/contentDecoding Da Vinci: Finance, functions and art
https://plus.maths.org/content/decoding-da-vinci
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Tim Johnson </div>
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<p>Dan Brown in his book, <em>The Da Vinci Code</em>, talks about the "divine proportion" as having a "fundamental role in nature". Brown's ideas are not completely without foundation, as the proportion crops up in the mathematics used to describe the formation of natural structures like snail's shells and plants, and even in Alan Turing's work on animal coats. But Dan Brown does not talk about mathematics, he talks about a number. What is so special about this number?</p>
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<div class="rightimage" style="width: 200px;"><img src="https://plus.maths.org/content/sites/plus.maths.org/files/articles/2011/davinci/fibonacci.jpg" alt="Fibonacci" width="200" height="270" /><p>Fibonacci (ca 1170 – ca 1250).</p><p><a href="https://plus.maths.org/content/decoding-da-vinci" target="_blank">read more</a></p>https://plus.maths.org/content/decoding-da-vinci#commentscontinued fractionFibonacciFibonacci numbergolden ratiomathematics and artpower seriesThu, 03 Nov 2011 12:58:56 +0000mf3445576 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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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/content