3
$\begingroup$

$$\sum_{n=1}^\infty z^{n!}$$

Here is what I've got so far

Claim: The above series converges for $|z|<1$.

Pick $|z|. Then for all $n$, $|z^{n!}|<=r^{n!}$.

So $\sum\limits_{n=1}^\infty r^{n!}$ is a majorant for $\sum\limits_{n=1}^\infty z^{n!}$.

$\sum\limits_{n=1}^\infty r^{n!}$ is a real series so we can test for convergence.

This is where I get stuck, I've tried the ratio test but that doesn't seem to work and I can't think of a function that would work for the comparison test.

  • 2
    Notice that $\sum_n r^{n!} < \sum_n r^n < \infty$ for $r <1$.2012-11-04
  • 0
    Aah cool so i can just use the comparison test on that. Thanks2012-11-04

4 Answers 4