convergence of a power series

**Veve** · August 22nd 2011, 04:05 AM

I have the following problem: Prove that the power series \sum \frac {z^n}{n} converges at every point of the unit circle except z=1.

The part with z=1 I already did it, but I have problems with the rest. Is there a way to prove the convergence using Fourier coefficients?

Thanks.

**girdav** · August 22nd 2011, 04:22 AM

Let $s_j:=\sum_{k=0}^jz^k$ . We have for $n\geq 1$ :
$\begin{align*}\sum_{k=1}^n\frac{z^k}k&=\sum_{k=1}^ n\frac{s_k-s_{k-1}}k\\&=\sum_{j=1}^n\frac{s_j}j-\sum_{j=0}^{n-1}\frac{s_j}{j+1}\\<br /> &=\frac{s_n}n+\sum_{j=1}^ns_j\left(\frac 1j-\frac 1{j+1}\right)\end{align*}$
Now show that for all integer $j$ we have $|s_j|\leq \frac 1{\sin\left(\frac{\theta}2\right)}$ .

**chisigma** · August 22nd 2011, 05:59 AM

Originally Posted by Veve

I have the following problem: Prove that the power series \sum \frac {z^n}{n} converges at every point of the unit circle except z=1.

The part with z=1 I already did it, but I have problems with the rest. Is there a way to prove the convergence using Fourier coefficients?

Thanks.

The Abel test for convergence of a series extablishes that...

a) given a sequence $a_{n}$ such that $\forall k$ the partial sum $s_{k}= \sum_{n=1}^{k} a_{n}$ is bounded...

b) given a sequence $b_{n}$ so that $|b_{n}|$ is decreasing and $\lim_{n \rightarrow \infty} b_{n}=0$ ...

... then the series $\sum_{n=1}^{\infty} a_{n}\ b_{n}$ converges. In Your case $b_{n}=\frac{1}{n}$ satisfies to criterion b), so that we have to verify thwe criterion a) for the sequence $a_{n}= e^{i n \theta}\, \ \theta \ne 0$ . The partial sum is...

$s_{k}= \sum_{n=1}^{k} e^{i n \theta} = e^{i \theta}\ \frac{1-e^{i k \theta}}{1-e^{i \theta}}$ (1)

... so that is...

$|s_{k}| = \sqrt{\frac{1-\cos k \theta}{1-\cos \theta}}$ (2)

... and for $\theta \ne 0$ it is evident that (2) is bounded. Then for Abel's test the series $\sum_{n=1}^{\infty} \frac{e^{i \theta}}{n}$ converges if $\theta \ne 0$ ...

Kind regards

$\chi$ $\sigma$

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