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Here's the exercise:

Let $\delta\in(0,1)$ and let $(a_n)_{n\in\mathbb{N}}$ be a real, monotonic decreasing sequence that converges to $0$. Show that $\sum a_nz^n$ converges uniformly on $\{|z|\leq1\}\cap\{|z-1|>\delta\}$

I quite frankly don't have any idea of how to approach this. Any hints and nudges in the correct direction are greatly appreciated.

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    Thanks, I edited that in. As for your problem, have you seen [Dirichlet's test](http://en.wikipedia.org/wiki/Dirichlet's_test)?2012-05-17

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Hints:

  • Use the fact that $(a_n)$ is decreasing and converges to zero to show that there exists a positive and summable sequence $(b_n)_n$ such that $a_n=\sum\limits_{k=n}^{+\infty}b_k$ for every $n$.
  • Use summation by parts to write $\sum\limits_{n\geqslant N}a_nz^n$ in terms of $(b_n)_{n\geqslant N}$ and $z$, for every $N$.
  • Show that $\left|\frac{z^i-z^k}{1-z}\right|\leqslant\frac2\delta$ uniformly on $z$ in the domain you consider, for every $i$ and $k$.
  • Deduce that $\left|\sum\limits_{n\geqslant N}a_nz^n\right|\leqslant\frac2\delta\,\sum\limits_{n\geqslant N}b_n$ for every $z$ in the domain you consider and for every $N$.
  • Conclude.
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    Wow. That is amazing.2012-05-17