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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?

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mauna
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  • The biggest problem with something like infinite series is that, even if they're convergent, it's rare that you'll be able to find a closed form. Some techniques include finding the limit of the sequence of partial sums, comparing with integrals, using techniques from analysis to estimate bounds, and so on. – Lost Dec 06 '13 at 13:16

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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.

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Old John
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    Generations of mathematicians have read that Knopp book. Unless you speak German, be glad that is had been translated... – GEdgar Dec 06 '13 at 14:15
  • I have used it many times, and still delve into it sometimes - luckily in English :) – Old John Dec 06 '13 at 17:20