How to determine the convergence of sequence
$$ \sum^\infty_{n=1} \tan^2(1/n) $$
I have used the divergence test
$$ \lim_{n\to\infty} \tan^2(1/n) = \tan^2(0) = 0 $$
So we can't say if it diverge or not. I know i should use a comparison test but i don't know how to choose the other sequence to compare it with.
Best Answer
$f(x)=\tan(x)$ is a non-negative convex function on the interval $[0,1]$, hence it follows that: $$ 0\leq \sum_{n=1}^{N}\tan^2\frac{1}{n}\leq\tan^2(1)\sum_{n=1}^{N}\frac{1}{n^2}\leq\frac{\pi^2\tan^2(1)}{6}.$$