Convergence/divergence of $\sum\limits_{n=1}^{\infty}\left(\cos\frac{1}{n}\right)^{n^3}$ How do I show convergence/divergence of the series
$$\sum\limits_{n=1}^{\infty}\left(\cos\frac{1}{n}\right)^{n^3}?$$
I begin by writing $\left(\cos\frac{1}{n}\right)^{n^3} = e^{n^3\ln\left(\cos\frac{1}{n}\right)}$ and continue by Taylor expanding around $0$; first cosine, then ln. But I get nowhere. I would appreciate any help.

A possible way to show convergence is to rewrite 
$$\cos \frac 1n = \cos \frac{2}{2n} = 1- 2\sin^2 \frac 1{2n} $$
and now use root test and the standard limits $\lim_{t\to 0}(1-t)^{\frac 1t} = \frac 1e$ and $\lim_{t\to 0}\frac{\sin t}{t}=1$:
\begin{eqnarray*}\sqrt[n]{\left(1- 2\sin^2 \frac 1{2n}\right)^{n^3}}
& = & \left(\left(1- 2\sin^2 \frac 1{2n}\right)^{\frac{1}{2\sin^2 \frac 1{2n}}}\right)^{n^2\cdot 2\sin^2 \frac 1{2n}}\\
& \stackrel{n\to \infty}{\longrightarrow} & \left(\frac 1e\right)^{\frac 12}=\frac 1{\sqrt e} < 1
\end{eqnarray*}
