Find the coefficient of $\frac{1}{n^4}$ in the given Summation.

Question

$$S=\sum_{r=1}^\infty \frac{(2^r)}{(r)(3n+2)^r}$$ $$\text{I began with writing S as } S=-log(1-\frac{2}{3n+2})$$ But further I fail to understand exactly how to extract the required term from the expansion.

Answer 1

In fact, you have $$S=\log \left(\frac{3 n+2}{3 n}\right)=\log\left(1+\frac{2}{3 n}\right)$$ Now use the expansion of $\log(1+x)$ in which you will make later $x=\frac{2}{3 n}$