I'm trying to find out whether $\sum _{n=0}^{\infty }\left(\cos^n\left(\frac{1}{\sqrt{n}}\right)-\frac{1}{\sqrt{e}}\right)$ converges or not. I've tried with taylor series but it doesn't lead me anywhere except with the fact that $\lim_{n \to \infty}\cos^n\left(\frac{1}{\sqrt{n}}\right)-\frac{1}{\sqrt{e}}=0$ and therefore it has "a chance" to converge.
Any hint?
Best Answer
hint
$$\cos(\frac{1}{\sqrt{n}})=1-\frac{1}{2n}+\frac{1}{24n^2}+o(\frac{1}{n^2})$$
$$\ln(1+\cos(\frac{1}{\sqrt{n}})-1)=$$
$$-\frac{1}{2n}-\frac{1}{12n^2}+o(1/n^2)$$
thus
$$\cos^n(\frac{1}{\sqrt{n}})=e^{n\ln(\cos(\frac{1}{\sqrt{n}}))}$$
$$=e^{-\frac 12}e^{-\frac 1n(\frac{1}{12}+o(1))}$$ $$=\frac{1}{\sqrt{e}}(1-\frac{1}{12n}(1+o(1))$$