How to find out whether the series $\sum\limits_{n=1}^{\infty}\left (\cos\frac{a}{n}\right )^{n^{3}}$ convergent or divergent?

Question

$$\sum_{n=1}^{\infty}\left (\cos\frac{a}{n}\right )^{n^{3}}$$ So I have such series. Firstly, I've tried Cauchy`s ratio test, but no results ( I have limit equal to 1). Then I've tried to write cosine as Taylor series up to 3rd term, but still nothing. Give me some hints,please.

Answer 1

$$\cos\frac an=1-\frac{a^2}{2n^2}+O(n^{-4})$$ so $$\ln\cos\frac an=-\frac{a^2}{2n^2}+O(n^{-4})$$ and $$n^3\ln\cos\frac an=-\frac{a^2n}{2}+O(n^{-1}).$$ Thus $$\left(\cos\frac an\right)^{n^3}=e^{-a^2n/2}\exp(1+O(n^{-1})).$$

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