# Math Help - Radius of convergence

How do I find R when the power series has n! as the power of x?

Thanks for helping out.

2. Originally Posted by DaRush19
How do I find R when the power series has n! as the power of x?
If you want help, you give the exact question.
There is no point in our guessing.

3. e.g Find the radius of convergence of the sum of 1/n! x^n!

4. For $|x| \le 1$ is $|x|^{n!} \le |x|^{n}$ so that...

$\sum_{n=0}^{\infty} \frac{|x|^{n!}}{n!} \le \sum_{n=1}^{\infty} \frac{|x|^{n}}{n!} = e^{|x|}$ (1)

... and the series of powers of x converges. If $|x|>1$ we set $|x|= 1 + \delta, \delta >0$ so that is...

$\frac{|x|^{n!}}{n!} = \prod_{k=1}^{n} \frac{(1+\delta)^{k}}{k}\rightarrow \ln (\frac{|x|^{n!}}{n!} ) = \ln (1+\delta) \sum _{k=1}^{n} k - \sum_{k=1}^{n} \ln k$ (2)

But is $\ln (1+\delta) >0$ so that is...

$\lim_{n \rightarrow \infty} \frac{|x|^{n!}}{n!} = + \infty$ (3)

... and the series of powers diverges. Then it seems to be $R=1$...

Kind regards

$\chi$ $\sigma$