What is the theoretical interest of finding closed-form solutions of infinite series?

Question

I was reading this when I came across Gourevitch's conjecture.

My understanding is that solutions to these series are of practical interest. If one encounters such a series, being able to solve it exactly is more practical than having to solve it numerically.

But, not being a mathematician, I simply can't imagine what the theoretical implications of proving such conjectures are.

What are they?

Answer 1

I can recommend reading "Closed forms: what they are and why we care" by Jon Borwein and Richard Crandall, the article is to appear in Notices Amer. Math. Soc. 60 (2013).

Edit (Dec 2012). The paper has just appeared in the Notices: pdf. Together with the sad news about Richard Crandall: he passed away.

Answer 2

Let $k \ge 2$ be an integer. Consider the series $$ \sum_{n=1}^{\infty} \frac{1}{n^k} $$ this is typically denoted $\zeta(k)$ as it is the value of the Riemann zeta function at $k$.

Now, Euler showed that for even $k$ this is equal to $$q_k \pi^k $$ where $q_k$ is some rational number (that one can also describe explictly).

This is on the one hand an interesting fact and would also allow to calculate approximations of the powers of $\pi$ but what I actually want to get at is that from this it follows that if one knows that $\pi$ is transcendental then one gets directly that $\zeta(k)$ is transcendental and in particular not a rational number.

So this is for even $k$. What about odd $k$? Say $k=3$. Is this rational or irrational? This question was open for a long time until it was proved at the end of the 1970s by Apéry.

How does this prove go (very roughly!):

He first showed that $$ \zeta(3) = \frac{5}{2} \sum_{i=1}^{\infty} \frac{ (-1)^n}{n^3 \binom{2n}{n}}.$$

So one could say he evaluated the series on the right in 'a closed form'; showing that its values is something already known/defined.

Then based on this he derived some sequences of rational numbers that converge to $\zeta(3)$ so fast that it is impossible for $\zeta(3)$ to be rational itself thus proving the irrationality of $\zeta(3)$.

So, in order to show that $\zeta(3)$ is irrational he first needed to show that it is equal to the limit of this (other) series, or put differently to evaluate this series; not only the first simpler one.

It would be interesting to be able to do something like this for other odd numbers, but so far no-one knows how to do so and the irrationality of $\zeta$ at any other odd positive integers is unknown (although there are results that assert that among certain collections of them there are at least some irrational ones).

Thus finding an evaluation of a series can be used to infer something theoretical on its value.

This is not always the motivation, but sometimes it is the case that the point is not so much to know the value of the series in order to replace it in some computation say, but rather to use the series as a form of describing its value by simpler building blocks and thereby allowing to learn something (new) on the value.

Answer 3

HERE is an example in MO. For the first question, I could evaluate the series in closed form, so I could compute with their known properties, and thus answer the question. For the second question, the series were just ${}_2F_1$ series, and I did not know their properties, so I could not answer the question. Despite the numerical evidence in the graphs.

Answer 4

Having the close form for a series of functions is also important:

1. Closed forms can sometimes be used to find exact solutions to differential equations.
2. Closed forms allow for accurate estimates on errors in approximations of special functions. Some of these are used in your calculator.
3. Closed forms can sometimes produce proofs of non-series results. For example, the equation $$ e^{ix} = \cos(x) + i\sin(x) $$ is proven by using three Taylor series.
4. This is a bit incestuous, but having the closed form for one series allows for manipulation to produce the closed form of other series.

Answer 5

This is an old question, but just popped up again, so I thought I'd mention a very important situation where explicit formulas are of fundamental theoretical importance. Number theorists are very interested in understanding the ideal class group $H_p$ of the cyclotomic field $\mathbb Q(\zeta_p)$, where $\zeta_p$ is a primitive (prime) root of unity. People use explicit formulas for special values of $L$-series to construct integers that annihilate the class group, thereby obtaining information about the size of the class group. And for elliptic curves, Rubin in the CM case and Kolyvagin for curves over $\mathbb Q$ have used special values such as $L(E,1)$ and $L'(E,1)$ to prove that the Tate-Shafarevich group of $E$ is finite, which had been a long-standing conjecture. Very (very) roughly, they use the special values to construct an integer $m$ that annihilates every element of SHA, and from there the finiteness follows, since it is not hard to prove that SHA$[m]$ is finite. This is the theory of Euler Systems, which is still attracting lots of research attention.

I realize that $L$-series are Dirichlet series, and this question asks about "infinite series", which is probably meant to refer to power series. But one can move back and forth between $L$-series and power series using the Mellin transform, and indeed, for elliptic curves over $\mathbb Q$, the first step in studying $L(E,1)$ is to write it as the Mellin transform of a modular form, courtesy of Wiles et al.