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).
Best Answer
No, there is no general formula yet to know the sum of
$$S(p)=\sum_{n=1}^{\infty} \frac{1}{n^p}$$
But we know that it converges iff $p>1$
For every $p$ even we know that:
$$S(p)=(-1)^{\frac p2+1}{{B_p(2\pi)^p}\over {2p!}}$$
where $B_n$ is a Bernoulli number.
However for odd integers we still don't have a general formula, if you are interested in this kind of series search about Riemann's Zeta function which is a very big topic in math.
More in general if you want to find the value of a random serie you have different convergence tests you can try (like Dirichlet's or Cauchy's one) and then you have to be smart and rearrange the terms in a way you can get out a result, like using telescopic series or Taylor expansions of some well-known functions.