Originally Posted by

**johng** Hi,

In your statement of an integral test, who is function f? Answer: f is a __decreasing__ non-negative function on say $[1,\infty)$ with $f(n)=a_n$ and $\int_1^\infty f(x)\,dx$ is finite. Here also $R_n=\sum_{k=n+1}^\infty a_n$.

Your problem:

$$b_n={sin^2(n)\over n^5}\text{. Then }R_n=|\sum_{k=n+1}^\infty b_n|\leq\sum_{k=n+1}^\infty {1\over k^5}$$

So let $f(x)={1\over x^5}$. Use this to find an error bound for $\sum_{k=n+1}^\infty {1\over k^5}$ and hence for $R_n$

Your second series is an alternating series with the terms decreasing.