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I think one reason for studying series is for power series. Are there any other application for studying series of numbers? Please let me know if you have any idea or comment for it.

Thanks in advance!

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    $\begingroup$ Possible duplicate of The Power of Taylor Series $\endgroup$ – Jam Apr 11 '18 at 12:42
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    $\begingroup$ I think this question is not a duplicate of The Power of Taylor Series, because that question is specifically about Taylor series whereas this question is asking about applications of series other than power series. I'm interested in this myself because I've thinking about how to motivate the study of infinite series when teaching calculus. $\endgroup$ – littleO Apr 11 '18 at 12:57
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    $\begingroup$ If only power series are important, then we can avoid Dirichlet series, and all those hard things in number theory! $\endgroup$ – GEdgar Apr 11 '18 at 13:13
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    $\begingroup$ "Most people" tend to think the value of a real number is a certain infinite series representation of it, such as $\sqrt{2} = 1.4142 \ldots = 1 + \frac{4}{10} + \frac{1}{100} + \frac{4}{1000} + \frac{2}{10000} + \cdots$ $\endgroup$ – Dave L. Renfro Apr 11 '18 at 14:50
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    $\begingroup$ Though not falling far from the power series tree, Fourier series are extremely useful. When considered as real functions, they are not power series. $\endgroup$ – Paul Sinclair Apr 11 '18 at 19:43
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There are lots of reasons one might wish to study infinite series. My introduction to infinite series involves a question: Can you determine $\sqrt{e}$ to, say, 5 decimal place accuracy without a calculator?

Basically, as a starting point for studying any kind of irrational quantity, we need some kind of infinite iterative process that will converge toward that quantity. Infinite series (and especially power series) provides a powerful tool in the analysis of the irrational.

Here are a few more useful things that can be done with infinite series:

  1. Certain integrals can be evaluated numerically using power series much more easily than using a Riemann Sum. For example: $\int\limits_0^1 e^{-x^2}dx$
  2. The definitions of $e^x,\sin(x),$ and $\cos(x)$ involving power series tend to be the most useful definitions for analytical purposes.
  3. Some difficult limits become downright obvious when the functions are expressed in terms of a power series. For example: $\lim\limits_{x\rightarrow 0}\frac{x\arctan(x^2)-x^3}{x^5}$
  4. The geometric series and binomial series are not only super useful for generating infinite series that add up to other things (e.g $\pi$, values of the natural log function, square roots, etc...), they are also of historical importance in the development of calculus.

Isaac Newton derived the power series for $\sin(x)$ in the following incredible way:

  • He used his binomial series to get the power series for $\frac{1}{\sqrt{1-x^2}}$
  • He then integrated this to get a series for $\arcsin(x)$
  • He then inverted this series to obtain the power series for $\sin(x)$, which required, in my opinion, a heroic amount of algebra.

This book has a nice treatment of how Newton achieved this.

  1. Summations are good practice for beginning programmers to write for loops.
  2. Most importantly, it's fun to think about how to add up an infinite number of numbers, or even determining whether adding up an infinite number of numbers will result in a number.
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Generating functions use series to solve certain counting problems by considering the coefficients of the series rather than the value of the series itself. You can even create generating functions which do not converge as a sum but the coefficients can be interpreted in a meaningful way. For example if I ask you how many ways you can make change for $n$ cents using coins of varying values like $1,5,10$ and $25$ cents you can solve this problem using a generating function.

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One reason is that series can provide an algorithm for computing particular functions. If I asked someone what $\sin(1)$ was, I wouldn't be surprised if they looked blankly and shrugged at me. But if I asked them what $1-\frac{1}{6}+\ldots$ was, they could easily figure it out and tell me.

Another reason is that an operator on a function (might) be easier to compute when the function is a series. For example, $\frac{\mathrm{d}}{\mathrm{d}x}f(x)=\frac{\mathrm{d}}{\mathrm{d}x}\sum a_ix^i=\sum \frac{\mathrm{d}}{\mathrm{d}x}a_ix^i=\ldots$.


Following on from @DaveL.Renfro's comment, since decimal expansions are infinite series in $10^i$, series can also give us a way of comparing the size of two objects. At a glance, it's hard so say if its true that $e<\pi$ but it's easier to show that $2+\frac7{10}+\frac1{100}+\ldots<3+\frac1{10}+\frac4{100}+\ldots$ i.e. $2.718\ldots<3.141\ldots$.

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    $\begingroup$ Thanks again. You also give the example related to power series ( or Laurent series). I want other application for series. $\endgroup$ – 04170706 Apr 11 '18 at 12:45
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Other comments have talked about uses of infinite series in mathematics. But they are also everywhere in applications. For example, in physics, if you want to compute the potential energy of an atom in a crystal, you have to take into account the contribution of the interaction of every other atoms. You end up with and infinite series. It is called Madelung constant.

Series are almost as ubiquitous as integrals. I can think of uses in statistical physics, optics (when you study reflexions on two crystals), biology (populations), and even philosophy (Zeno's paradox).

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    $\begingroup$ Nice reference to Madelung. Also worth noting that the original Madelung series have some trouble with convergence, and computing Madelung constants with any kind of accuracy wasn't possible until some great work done by mathematicians $\endgroup$ – Yuriy S Apr 12 '18 at 7:43
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In number theory there are other important kinds of series. For example, greedy algorithm and Engel expansion allow us to represent both rational and irrational number as a sum of unit fractions.

Both of those expansions can be used to prove or disprove irrationality, because they are finite if and only if the number is rational.

If the number is irrational, its expansion would be infinite, and can be seen as an infinite series.

Moreover, for some numbers, like algebraic integers, greedy expansion has a pattern, which can be used to compute their digits:

$$3-2 \sqrt{2}=\frac{1}{6}+\frac{1}{204}+\frac{1}{235416}+\dots$$

Denominators are $2^n$th terms of the recurrence $A_n=34A_{n-1}-A_{n-2},~A_0=0,~A_1=6$. http://oeis.org/A082405

$$4-2 \sqrt{3}=\frac{1}{2}+\frac{1}{28}+\frac{1}{5432}+\dots$$

Denominators are $2^n$th terms of the recurrence $A_n=14A_{n-1}-A_{n-2},~A_0=0,~A_1=2$. http://oeis.org/A011944

The rest can be seen in a question of mine (with an excellent answer given by Noam D. Elkies):

Numbers $p-\sqrt{q}$ having regular egyptian fraction expansions?

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