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

Question

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

freckles · Answer

*

zarkon · Answer

$$\tan(1)$$

anonymous · Answer

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

anonymous · Answer

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 $\tan(1)$

anonymous · Answer

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.

zarkon · Answer

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