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All calculus textbooks I know of seem to be obsessed with the question of which infinite series is convergent and which is not but none address the question of how to find the sum of an infinite convergent series . What books contain systematic treatment of methods of finding sums of series ?

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Finding the sum of a convergent series is generally a next-to-impossible task. Consider the (apparently relatively simple) case of $\sum_{n=1}^\infty n^{-r}$ for $r>1$. –  Andrea Mori Dec 19 '12 at 10:58
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A good deal of Graham, Knuth, & Patashni, Concrete Mathematics, is devoted to methods for solving this very difficult problem in certain cases. –  Brian M. Scott Dec 19 '12 at 11:10

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The best book I know of that treats Infinite series is Theory and Application of Infinite Series by Konrad Knopp. It is very complete and takes some effort to get through, but if you absorb the lessons he teaches, you will be as well-equipped as anyone here to answer questions about infinite series.

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There is also the famous book by George Polya : Problems and Theorems in Analysis I: Series, Integral Calculus, Theory of Functions.

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Although it will be a book of exercises and their solutions contains general methods to find out if the series. It is one of the references of Knopp.

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