## anonymous one year ago Can we find a closed form? $\lim_{k\to\infty}\sum_{n=1}^\infty \tan\frac{1}{n^k}$ Numerically, the limit approaches $$1.5574078$$.

1. freckles

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2. Zarkon

$\tan(1)$

3. anonymous

Hmm, for $$n>1$$ terms, you have $$\tan(0) = 0$$.

4. anonymous

the behavior is a consequence of the fact that $$1/n^k\to 0$$ very quickly as $$k\to\infty$$ for $$n>1$$, so quickly in fact that we can observe that only the $$n=1$$ term will 'survive' and we get $$1/1^k=1$$ and thus we get $$\tan(1)$$

5. anonymous

Right, this question seemed much more difficult at first glance when I was typing random series into Mathematica :P I'll try to find something more challenging.

6. Zarkon

you can't just bring the limit past the sum...you need justification for that