Some nice theorem to find the value of serie 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 !
 A: 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
A: I don't know if this is a good approach and if you mentioned this but it would add another possible approach which I posted on the linked question.
Consider
$$
F(y+1)-F(y) = e^{i\pi y} x^{y^2} = e^{i \pi y -ay^2} = f,
$$
Take fourier transform of this and solve e.g.
$$
\hat F(s) = \frac{\hat f(s)}{e^{is} - 1}
$$
 Note that
$$
\lim_{n \to \infty} \sum_{i=0}^{n} F(i+1)-F(i) = - F(0) = \lim_{n \to \infty} F(n+1) - c \int \mathcal{F}(f)(s)\,ds = - c \int \mathcal{F}(f)(s)\,ds,
$$ under the assumption $\lim_{n\to\infty} F(n+1) = 0$
For the example above it turns out the fourier transform turns into a delta measure as we approach $x\to 1$ and this might be an approach that is tractable and lead to convergence proof and limit calculation.
