TITLE: Which (convergent) series can one find the sum of?
QUESTION [5 upvotes]: I know about geometric series and how one can find the sum when they are convergent.
I also have heard that one can prove that the $p$-series 
$$\sum_{n=1}^{\infty} \frac{1}{n^2}$$
has sum $\pi^2/6$ (but I don't know how this is actually proved).
This might be an unanswerable question, but are there any tools or general rules that will tell you when you can find the exact sum of a given series? Are there, for example, rules/tools for how to find the exact value of any $p$-series? (like with geometric series).

REPLY [2 votes]: It is, in general, very hard to prove there is NO closed form expression for the sum of a series when the terms of the series are given by a formula. It is much easier, for example, to prove that a closed form formula function has no closed form formula anti-derivative.
On the other hand, there is no reason why we shouldn't be able to (eventually) come up with a closed-form expression for many series sum for many series that we currently can't. This is because both the collection of series whose terms can be expressed in closed-form formula has the same "size" (namely countable) as the number of series whose sum can be expressed in closed form expression. On the other hand, there are uncountably many (i.e. more than countable) convergent series even if we restrict terms to be rational.
When binomial coefficients and such are involved in terms of a series, and also in some other cases, hyper-geometric expressions can often offer a closed-form solution (if you accept that a hyper-geometric expression is a closed form). Manipulations of such series to arrive at closed-form expression for the answer are often tedious, so tools like wolphram alpha are often convenient.