# What is the importance of knowing if a series converges or diverges? [closed]

This semester I have been given a multitude of techniques for discovering if a series converges or diverges with no explanation for why I would need to know this. It would be helpful if I could better understand the importance of knowing if a series converges or diverges. How would this be used in a non-classroom setting?

## closed as off-topic by Will Jagy, 3SAT, John B, Shailesh, Daniel W. FarlowMar 11 '16 at 0:36

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• "This question is not about mathematics, within the scope defined in the help center." – Will Jagy, 3SAT, John B, Shailesh, Daniel W. Farlow
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• This is a very strange way of asking the question. If you want to know what infinite series are for, you could ask that. But once given that it makes sense in some way to know about infinite series, how could it not matter whether it converges? I'm tempted to say that if it doesn't converge, it has no value. But that would be a pun, so I won't say that. $\qquad$ – Michael Hardy Mar 10 '16 at 20:47
• @MichaelHardy Besides being a pun, that would be wrong. Asymptotic series are often useful, despite the fact that they don't converge anywhere except $0$. – Robert Israel Mar 10 '16 at 21:21
• The notion of convergence is useful in other math subjects like topology: you define open and closed sets based on if the limits of a convergent series (Cauchy series) are in the set or not. – Broken_Window Mar 10 '16 at 22:53
• @RobertIsrael : True, but I still think it's an odd way to ask a question. $\qquad$ – Michael Hardy Mar 10 '16 at 23:34
• If you are running a computer program to solve a problem, and the series consisting of the sequence whose elements are computed at each iteration of the algorithm converges, then the problem is solved. Otherwise, the program will never stop and your computer's memory will run out. See this. This is quite important in complexity analysis in computer science! cse.sc.edu/~fenner/csce750/course-notes.pdf – Carlos - the Mongoose - Danger Mar 11 '16 at 0:37

## 5 Answers

Without convergence, an infinite series has no value (well, at the level you're working at this is true). Therefore your question is essentially the same as "if I know whether or not an infinite series is meaningful, why does it matter?"

The main reason you're learning about infinite series is because later on in your course you will meet power series, which allow you to approximate complicated functions by the simplest functions of all: polynomials. Those convergence tests help you determine where power series make sense. Infinite series are also used to define new functions (e.g., Bessel functions, zeta-functions, elliptic functions), to extend the meaning of familiar functions to new settings (e.g., the exponential or sine of a matrix or of a complex number), and to solve differential equations (series solutions). This last point was Newton's great discovery: he thought he could solve "any" problem in math by writing the answer as an infinite series and then figuring out what the coefficients had to be to satisfy the problem.

One obvious use: if you know a series converges, say $$S= \sum_{k=0}^{\infty}a_k$$ then you can approximate $S$ as closely as you like by just adding up a finite number of terms:$$S\approx \sum_{k=0}^{N}a_k$$ You can't really "add" the terms in an infinite series -- you can only take a limit of partial sums. But if the series converges, even if you can't find the limit analytically, you can approximate it numerically by simple addition.

• Why does this explain the importance of knowing whether a series converges? It doesn't describe any application to a problem not already about infinite series themselves. It's like saying convergence of a sequence can be used to estimate the limit. That doesn't really address why you'd want to do it in the first place. – KCd Mar 11 '16 at 0:08

Series have a lot of applications in many fields of mathematics: from the definition of Euler's constant as $\sum_{n=0}^{\infty}\frac{1}{n!}$ to the convergence of a series of functions to another function. Generally, many things can be defined trought series therefore knowing how they work is very important. Just know that you won't have to wait long to apply them in your studies. Last, a little example of an application to a famous paradox.

PS a general note: many things in mathematics were born just because of someone's broad fantasy and seemed to have no application. This could hold true for some time, untill, hundreds of years later someone found an application, like non-euclidean geometry and Einstein's relativity.

• "$e$" is not not the Euler constant many people would think of... Maybe better to just call it "basis for the natural logarithms" or something. – paul garrett Mar 11 '16 at 0:29

Many equations don't have "closed form" solutions, but often you can write a solution in the form of an infinite series. In order to actually use that series solution, you will want to approximate it numerically, typically by taking the sum of a finite number of terms and using that as your approximation. Now the question is, can this really give you a good approximation to an actual solution? If the series converges to a solution, then the answer is yes: you can get as good an approximation as you want to that solution by taking the sum of enough terms of the series.

• If you don't know what the series converges to (can't actually find the limit), how do you find out how good your approximation is, how "far" you are from the actual limit value? – jeremy radcliff Mar 11 '16 at 0:33
• There are techniques to estimate the tail of a series. – Robert Israel Mar 11 '16 at 2:31

Convergence of series is exactly what is needed to put trigonometry on rigorous footings. The cosine function is defined as an infinite series.

$\cos(x) = \displaystyle \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}$

The series converging means essentially that the sum of infinitely many terms "makes sense". And of course we would like trig functions to make sense.

Many facts about trigonometry are much easier to prove sing the series definition rather than the (formally inadequate) definition involving circles. For example the identities

$\displaystyle \frac{\partial }{\partial x} \sin(x) = \cos(x)$

and

$\displaystyle \frac{\partial }{\partial x} \cos(x) = -\sin(x)$

and

$e^{i \pi} +1 =0$

fall out of the series definition very quickly and rigorously.

• For $x\in(0,\pi/2)$, and hence for all real arguments, it is possible to define trigonometric functions fully formally in a "school" way using a circle. What we need is a notion of length of a curve there, and this can be done without infinite series. – Wojowu Mar 10 '16 at 20:50
• This is incorrect. Using power series to define sine and cosine is used to extend their definitions to the complex numbers. – John Douma Mar 10 '16 at 20:53
• What is incorrect? – Daron Mar 10 '16 at 20:55
• You do not need infinite series to rigorously define the sine and cosine. – John Douma Mar 10 '16 at 20:56
• @Wojowu Of course, you are correct but at the same time if you rigorously follow through with the approach you suggest, you will need to define arc length and then at some point invoke the intermediate value theorem. Arguably none of this is quite elementary although I'd say it's much more intuitively clean than using infinite series per se. – Chan-Ho Suh Mar 11 '16 at 0:34