This is a question from a homework assignment that I turned in today and it's driving me nuts because I don't know if I did it correctly or not. First of all we have these two questions that seem almost identical:(adsbygoogle = window.adsbygoogle || []).push({});

Question 1

Show that the following series converges uniformly in the given region.

[tex]\displaystyle\sum_{n=1}^{\infty}z^{n}[/tex] for [tex]0\le|z|<R[/tex] and [tex]R<1[/tex]

Question 2

Show that the sequence:

[tex]\displaystyle\left\{x^{n}\right\}^{\infty}_{n=1}[/tex]

converges uniformly inside [tex]0\le|z|<R[/tex] with [tex]R<1[/tex].

How I did question 1:

[tex]\displaystyle\sum_{n=1}^{\infty}z^{n}= z + z^{2} + z^{3} + \cdots + z^{n} + \cdots[/tex]

Call the nth partial sum of the series [tex]S_{n}[/tex]:

[tex]S_{n}= z + z^{2} + z^{3} + \cdots + z^{n}[/tex]

[tex]zS_{n}= z^{2} + z^{3} + \cdots + z^{n} + z^{n+1}[/tex]

[tex]S_{n}=\frac{z(1 - z^{n})}{1-z}[/tex]

Now,

[tex]\displaystyle\sum_{n=1}^{\infty}z^{n}= \displaystyle\lim_{n\to\infty}S_{n}=\displaystyle\lim_{n\to\infty}\frac{z(1 - z^{n})}{1-z}[/tex]

[tex]=\frac{z}{1-z}[/tex] because [tex]z^{n}\longrightarrow 0[/tex] for [tex]|z|<1[/tex]

Now we consider:

[tex]\left|\frac{z}{1-z} -\frac{z(1 - z^{n})}{1-z}\right| = \frac{|z|^{n}}{|1-z|}< \frac{R^{n}}{1-R}[/tex]

with [tex]\epsilon>0[/tex] we choose N so that N depends on R and [tex]\epsilon[/tex] but not on on z, so that when [tex]n>N_{R,\epsilon}[/tex]

[tex]\frac{R^{n}}{1-R}< \epsilon[/tex]

Because we can choose such an N the series converges uniformly.

For question 2 I was confused, If

[tex]\displaystyle\sum_{n=1}^{\infty}z^{n}[/tex]

converges then the sequence [tex]\displaystyle\left\{x^{n}\right\}^{\infty}_{n=1}[/tex]

must be cauchy, right? So as [tex]n\longrightarrow \infty[/tex] the nth term of the series goes to zero. So I thought I'd need to show that there was a N, not dependent on z such that the nth term would be less than [tex]\epsilon[/tex].

I found:

[tex]N = \frac{log \epsilon}{log R} + 1[/tex]

Is that good enough to show uniform convergence?

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# Homework Help: Complex series question

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