Want to check analyticity of a series on a open disk.

Question

How do we check the analyticity of a any power series? For example:

How will we show that $$f(z):= \sum_{n=1}^\infty z^{n!}= z^1+z^2+z^6+z^{24}......+z^{n!}......$$ is anaytic on disk {$z : |z|<1$}

Thanks.

Answer 1

The coefficients $a_n$ of your power series are $0$ or $1$, and $a_n=1$ for infinitely many $n$, so $\limsup_{n\to\infty}|a_n|^{1/n}=1$. Thus, your power series converges on the open unit disk by Cauchy-Hadamard theorem, and by standard results it is analytic on such disk.