Thread: Find the sum of the serie explicitly

1. Find the sum of the serie explicitly

$\displaystyle |\omega|\leq 1, \omega\neq1$ 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

2. Originally Posted by KZA459
$\displaystyle |\omega|\leq 1, \omega\neq1$ 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

3. Thanks a lot and yes it was from 1 to infinity