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I need help showing the convergence of the sum $\sum_{n=0}^\infty a_nz^n$ on the unit circle (apart from z=1) when $a_n$ is a decreasing series of real numbers such that $\lim\limits_{n\to\infty} a_n=0$ and $\sum_{n=0}^\infty a_n$ diverges.

Im kind of stuck and will very much appreciate the help.

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    See [e.g. here](http://math.stackexchange.com/questions/118686/determining-precisely-where-sum-n-1-infty-fracznn-converges) (the step from $a_n = \frac{1}{n}$ to general decreasing $a_n \to 0$ is easy). – 2017-01-23

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