Trigonometric Series

How to show that $$\sum_{k=1}^{\infty} {\arctan{(1/k^2)}}$$ converges? I would prefer to avoid the integral test.

Can I find a more notorious convergent series that limit mine from above?

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Presumably $k$ and $n$ are somehow related? – GEdgar Jan 31 '13 at 22:22

Remember that $\lvert\arctan x\rvert \le |x|$...