Convergence of $\sum_{n=1}^{\infty}\left(\cos\frac{1}{n}\right)^{k(n)}\,\,\,\,; k(n)=\frac{1}{\sin^a\left(\frac{1}{n}\right)}$

Question

The convergence of the following series as $a \in \mathbb{R}$

$$\sum_{n=1}^{\infty}\left(\cos\frac{1}{n}\right)^{k(n)}\,\,\,\,; k(n)=\frac{1}{\sin^a\left(\frac{1}{n}\right)}$$

As $n \to +\infty$ we have that $\cos\frac{1}{n} \sim 1-\frac{1}{2n^2}$ and that $\sin^a\left(\frac{1}{n}\right) \sim \frac{1}{n^a}$ but I can't figure out how to handle these results. Better to use comparision test?

Answer 1

First you can have: $\left(\cos\frac{1}{n}\right)^{\frac{1}{\sin^a\left(\frac{1}{n}\right)}}=e^{\frac{1}{\sin^a\left(\frac{1}{n}\right)}\log\cos\frac{1}{n}}.$

And $\frac{1}{\sin^a\left(\frac{1}{n}\right)}\log\cos\frac{1}{n}=\frac{1}{\sin^a\left(\frac{1}{n}\right)}\log(1+\cos\frac{1}{n}-1)\sim\frac{\cos\frac{1}{n}-1}{\frac{1}{n^a}}\sim\frac{\frac{-1}{2n^2}}{\frac{1}{n^a}}=\frac{-1}{2n^{2-a}},$ as $n\to\infty.$ So $$\left(\cos\frac{1}{n}\right)^{\frac{1}{\sin^a\left(\frac{1}{n}\right)}}\sim e^{\frac{-1}{2}n^{a-2}}.$$ Base on this, you can give the convergence.

Answer 2

HINT

We have that

$$\cos\frac{1}{n} \sim 1-\frac{1}{2n^2}$$

$$\sin^a\left(\frac{1}{n}\right) \sim \frac{1}{n^a}$$

therefore

$$\left(\cos\frac{1}{n}\right)^{k(n)}=e^{k(n)\log \left(\cos\frac{1}{n}\right)}\sim e^{-\frac12 n^{a-2}}$$

and the given series seems to converge for $a>2$.

Now you can try to make it more rigorous taking into account the remainders and referring to limit comparison test.