How to find the radius of convergence and interval of convergence of the following power series?
$\sum{x^{n!}}$
and
$ \sum{\frac{1}{n^{\sqrt{n}}} x^n} $
How to find the radius of convergence and interval of convergence of the following power series?
$\sum{x^{n!}}$
and
$ \sum{\frac{1}{n^{\sqrt{n}}} x^n} $
For the first one just use the ratio test:
$\frac{x^{(n+1)!}}{x^{n!}}=x^{(n+1)!-n!}=x^{n!((n+1)-1)}=x^{n\cdot n!}\;.$
Now what’s $\lim_{n\to\infty}|x|^{n\cdot n!}\;?$
Hint: Use the root test for the second series and note that
$ \lim_{n\to \infty} \left(\frac{1}{n^{\sqrt{n}}}\right)^{1/n}=1. $