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I have calculated using the root test that the radius of convergence of $\displaystyle\sum_{n=0}^\infty z^{n!}$ is $1$.

But how would I show that there is an infinite number of $z \in \mathbb{C}$ with $|z|=1$ for which the series diverge? I don't really understand what is meant in the above question, could someone explain to me what the question means?

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    The more interesting question is what might happen as you move out, within the unit disk, towards a point on the boundary, and evaluate the function on such points.2012-05-26
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    Thanks Lubin, I'm currently thinking over this question, hope to make some progress soon.2012-05-26
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    This is a lacunary function: http://en.wikipedia.org/wiki/Lacunary_function2012-05-26

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