Finding an upper bound for the approximation error of a series. Suppose that $$s=\sum_{n=1}^{\infty}\frac{\cos^2(n)}{n^5+2n^2}$$ and $$s_n=\sum_{i=1}^{n}\frac{\cos^2(i)}{i^5+2i^2}$$, then the approximation error is defined as $R_n=s-s_n$. I need to find a higher bound for this error and also find a $n$ for which the approximation error is less than $10^{-8}$, to do this, I have tried the next:
$\frac{\cos^2(n)}{n^5+2n^2}\leq \frac{1}{n^5+2n^2}<\frac{1}{3n^2}$, so $\sum_{n=1}^{\infty}\frac{\cos^2(n)}{n^5+2n^2}<\sum_{n=1}^{\infty}\frac{1}{3n^2}<\infty$ therefore the series converges and as $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$, then I have to $s<\frac{\pi^2}{18}$ and so $R_n=s-s_n<\frac{\pi^2}{18}$ so $R_n<\frac{\pi^2}{18}$.
The problem is that this bound is not very good because to solve the next part of the problem I cannot do it, how can I limit this better? or how can I calculate this $n$ with this bound so that $R_n<10^{-8}$? Thank you.

Note that
$$|s-s_n|=\sum_{k=n+1}^\infty\frac{\cos^2(k)}{k^5+2k^2}<\sum_{k=n+1}^\infty\frac1{k^5}<\int_n^\infty\frac{\mathrm dx}{x^5}=\frac1{4n^4}$$
so it suffices to take $n\ge71$ to get $|s-s_n|<10^{-8}$.
