TITLE: What is the theoretical interest of finding closed-form solutions of infinite series?
QUESTION [9 upvotes]: 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?

REPLY [2 votes]: 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.