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How to justify the convergence and calculate the sum of the series: $$\sum_{n=1}^{+\infty}\frac{1}{1^2+2^2+\cdots+n^2}.$$

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Can you compare it to something else that you know converges? In particular, is there something bigger than it that converges? Also, the singular of "series" is still "series". – Christopher A. Wong Feb 28 '13 at 21:28
compare it with $\sum_{n=1}^\infty \frac{1}{n(n+1)}$ – user59671 Feb 28 '13 at 21:29
up vote 33 down vote accepted

$$\begin{array}{lcl} \sum_{n=1}^\infty \frac{1}{1^2+2^2+\cdots+n^2}&=& \sum_{n=1}^\infty\frac{6}{n(n+1)(2n+1)} \\ &=& 6\sum_{n=1}^\infty \frac{1}{2n+1} \left( \frac{1}{n}-\frac{1}{n+1}\right) \\ &=& 12\sum_{n=1}^\infty \frac{1}{2n(2n+1)} -12\sum_{n=1}^\infty \frac{1}{(2n+1)(2n+2)} \\ &=& 12\sum_{n=1}^\infty \left[ \frac{1}{2n}-\frac{1}{2n+1} \right] - 12\sum_{n=1}^\infty \left[ \frac{1}{2n+1}-\frac{1}{2n+2} \right]\\ &=& 12(1-\ln 2)- 12\left(\ln 2-\frac{1}{2}\right)\\ &=& 18-24\ln 2 \end{array} $$

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great thanks for the hint. – user45099 Feb 28 '13 at 21:53
Okay, how doesn't this converge to zero? The denominator grows without bound! – PyRulez Mar 2 '13 at 14:47
@PyRulez The sequence of terms in the sum converges to 0. The series doesn't. – Solomonoff's Secret Dec 5 '14 at 17:23

For the convergence use comparism with another sum.

Hint: $$\sum_{i=1}^n i^2 =\frac{n (n+1) (2n+1)}{6}$$ and use partial fraction decomposition.

As you know that the convergence is absolut you can change the summation order. (And that is important here).

Maybe another hint is $$\sum_{i=1}^\infty (-1)^i \frac{1}{i}=-\ln(2)$$ This is a result from the Taylor series of the logarithm

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