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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3$\begingroup$ 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". $\endgroup$– Christopher A. WongFeb 28, 2013 at 21:28
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1$\begingroup$ compare it with $\sum_{n=1}^\infty \frac{1}{n(n+1)}$ $\endgroup$– user59671Feb 28, 2013 at 21:29
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2 Answers
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$$\begin{array}{lcl} \sum_{n=1}^\infty \frac{1}{1^2+2^2+\cdots+n^2}&=& \sum_{n=1}^\infty\frac{6}{n(n+1)(2n+1)} \\ &=& 6\sum_{n=1}^\infty \frac{1}{2n+1} \left( \frac{1}{n}-\frac{1}{n+1}\right) \\ &=& 12\sum_{n=1}^\infty \frac{1}{2n(2n+1)} -12\sum_{n=1}^\infty \frac{1}{(2n+1)(2n+2)} \\ &=& 12\sum_{n=1}^\infty \left[ \frac{1}{2n}-\frac{1}{2n+1} \right] - 12\sum_{n=1}^\infty \left[ \frac{1}{2n+1}-\frac{1}{2n+2} \right]\\ &=& 12(1-\ln 2)- 12\left(\ln 2-\frac{1}{2}\right)\\ &=& 18-24\ln 2 \end{array} $$
answered Feb 28, 2013 at 21:38
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$\begingroup$ great thanks for the hint. $\endgroup$– user45099Feb 28, 2013 at 21:53
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$\begingroup$ Okay, how doesn't this converge to zero? The denominator grows without bound! $\endgroup$– PyRulezMar 2, 2013 at 14:47
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1$\begingroup$ @PyRulez The sequence of terms in the sum converges to 0. The series doesn't. $\endgroup$ Dec 5, 2014 at 17:23
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For the convergence use a comparison with another sum.
Hint: $$\sum_{i=1}^n i^2 =\frac{n (n+1) (2n+1)}{6}$$ and use partial fraction decomposition.
Since you know that the convergence is absolute, 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
answered Feb 28, 2013 at 21:30