I know that the series given converges by using the series ratio test, but how would I find the exact value?
$\sum _{n=2}^{\infty }\:\frac{2^{n+1}}{\left(n+1\right)3^n}$
The hint given was: use geometric series to write $-ln(1-x)$ as a power series. Any help to get started would be great.
Getting started: Recall that $$ \int_0^x\frac{1}{1-t}\mathrm dt=-\ln(1-x) $$ for $|x|<1$. Recall also that $$ \frac{1}{1-x}=\sum_{n=0}^\infty x^n $$ for $|x|<1$. So, putting it together $$ -\ln(1-x)=\sum_{n=0}^\infty \int_0^xt^n\mathrm dt=\sum_{n=0}^\infty \frac{x^{n+1}}{n+1} $$ and....