Prove series convergence $\sum_{n=1}^{\infty}{\frac{\cos n}{\sin 2^{-n}+n^2}}$

analysisconvergence-divergencesequences-and-series

I have series (1):
$$
\sum_{n=1}^{\infty}{\frac{\cos n}{\sin 2^{-n}+n^2}}
$$
I have to check this series to be convergent or not.

I suppose that it could be comparable with series (2):
$$
\sum_{n=1}^{\infty}{\frac{1}{n^2}}
$$
that is know to be convergent. I tried to prove it by using some comparison tests:

  1. Limit comparison test – I can't calculate resulting limit
  2. Others (integral, ratio, etc) are not applicable for these series.

My questions are:

  1. Is series (1) comparable with series (2) and if yes how to prove it?
  2. Is there any other way to check convergence/divergence of series (1)?

Best Answer

You can apply the comparison test to check absolute convergence:

\begin{align*} \left|\frac{\cos(n)}{\sin(2^{-n}) + n^{2}}\right| \leq \left|\frac{\cos(n)}{n^{2}}\right| \leq \frac{1}{n^{2}} \end{align*}

That is because $0 < \sin(2^{-n}) < 1$ for every $n\in\mathbb{N}$ and $|\cos(n)| \leq 1$.

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