# Tagged Questions

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### Show that $\sum_{n=1}^{\infty}z^{n!}$ diverges for infinitely many $z$ with $|z|=1$

Problem: I need to show that the power series $\sum_{n=1}^{\infty}z^{n!}$ diverges for infinitely many $z$ with $|z|=1$. I tried to prove it by contradiction by assuming that diverges for finitely ...
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### Radius of Convergence of $\sum\ z^{n!}$

Does anyone know how to find the radius of convergence of the series $\sum\ z^{n!}$, where $z$ is a complex number? I tried to use the definition: $\frac{1}{R}=Limsup|\frac{a_n+1}{a_n}|$, but I ...
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### How to construct this Laurent series?

How do I construct the following Laurent series (clipped off Wolfram Alpha)? I know that the numerator can be written as $-1+\frac{\pi}2 z-...$ Alternatively (without the Laurent series), how can I ...
### Determine if series are convergent in $(C([0,1]),\| \cdot\|_{\infty})$
Determine whether the following series $\sum_{n=1}^\infty f_n$ are convergent in the space $(C([0,1]),\|\cdot\|_\infty)$, where (i) $f_n(t)= \frac{t^n}{n!}$; (ii) $f_n(t)=\frac{t^n}{2n}$ ...