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Recently learned about the monotone convergence theorem.

I have the sequence: $x_n = \frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{n^2}$

I need help proving that it is increasing and bounded, hence converges by the monotone convergence theorem.

I can see that it is monotonically increasing and it's bounded below by 1 but I'm not sure how to prove these properties. Also how do I determine and prove the upper bound for the series?

Thanks in advance.

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  • $\begingroup$ Technically we could do $|x_n| \leq \frac{\pi^2}{6}$, if you are okaying with using $x_n \to \frac{\pi^2}{6}$ $\endgroup$ – IAmNoOne Oct 29 '14 at 2:09
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$\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}<\frac{1}{1*2}+\frac{1}{2*3}+...+\frac{1}{(n-1)*n}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{n-1}-\frac{1}{n}<1 $, so $ x_n<2 $ for all $ n $.

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