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
$$ \lim_{n\to \infty} \left(\frac{1}{n^{\sqrt{n}}}\right)^{1/n}=1. $$