What is the importance of knowing if a series converges or diverges?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.