Convergence of a Complex Power Series

**roninpro** · November 17th 2010, 07:36 AM

Hello everyone. I am asked to show that the power series

$\displaystyle \sum_{n=1}^\infty \frac{z^n}{n}$

converges for $|z|\leq 1$ , except for $z=1$ .

It is easy to show that the series converges to $|z|<1$ by the root test. However, it is not clear to me why it should converge on the boundary. I tried putting $z=e^{i\theta}$ , but it does not help.

I would appreciate any suggestions.

**chisigma** · November 17th 2010, 07:47 AM

See...

Abel's test - Wikipedia, the free encyclopedia

Kind regards

$\chi$ $\sigma$

Math Help - Convergence of a Complex Power Series

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