Given a series, what are the techniques to find a formula that sums the series to infinity?
I only know the method of multiplying the series with a factor and then taking their difference (like here). But today, I found out that we can also try to find such a formula by differentiation (see answer to this question).
Therefore, I was wondering what other techniques exist and if there is a book/online resource that I can read to understand the logic behind them?
Best Answer
There are many techniques, but unfortunately there is no general method which is guaranteed to succeed, hence the difficulty (perhaps impossibility) of finding things like a closed formula for $\zeta(3)$.
The techniques that I can recall using in the past include:
Using known power series expansions (including things like geometric series)
Differentiating or integrating power series
Using complex analysis (look for summation of series by using residues) as in this question
Fourier expansions, including Parseval's theorem - as in this question
However, there is no method which is always going to work, so it means lots of practice to gain experience!
I am sure others will be able to suggest more modern books, but one I have is:
Konrad Knopp - "Theory and Application of Infinite Series" (Dover edition) which has a whole chapter on closed and numerical expressions for sums of series.