TITLE: Some nice theorem to find the value of serie
QUESTION [6 upvotes]: I know a lot of techniques to show that a serie converge or diverge (Cauchy condensation test, Abel's test, comparision with an integrale, root test, ratio test...).
On the other hand I know nothing about caculating the value of a serie. Is there any theorems that may help ? 
From now on I know the following : 

integrale test can be combine with squeeze theorem
geometric serie 

For example in this [How to prove $\lim \limits_{x \to 1^-} \sum\limits_{n=0}^\infty (-1)^nx^{n²} = \frac{1}{2} \ $? there are talking about : 

Herglotz' trick
Weierstrass products.
...

So I would like to know if there are some other nice theorem or slick techniques to find the value of a serie. 
Thank you !

REPLY [2 votes]: In this regard, series are much like integrals: you need knowledge and, even more, ingenuity and experience.
Some basic Theorems, besides those you cited, are:
 - Indefinite Sum;
 - Summation by parts;
 - (Discrete) Convolution;
 - Partial Fractions Decomposition;
 - Binomial Tranform and others;
 - Gosper's Algorithm and Hypergeometric Function;
 - Abel's Binomial Theorem
and many others