$\displaystyle |\omega|\leq 1, \omega\neq1$ and $\displaystyle \omega$ is complex
$\displaystyle \Sigma_{n=0}^\infty \frac{\omega^n}{n}$
Find the sum of this serie explicitly
Any ideas would be appreciated, thanks
First I really hope that n=1 is where it starts or we have a problem...
With that said notice that if
$\displaystyle f(\omega)=\sum_{n=1}^\infty \frac{\omega^n}{n}$
we can take the derivative both sides to get
$\displaystyle f'(\omega)=\sum_{n=1}^\infty \omega^{n-1}$
If we reindex the series we get
$\displaystyle f'(\omega)=\sum_{n=1}^\infty \omega^{n-1}=\sum_{n=0}^{\infty}w^{n}=\frac{1}{1-\omega}$
Just integrate from here and you are done