Compute the radius of convergence of the power series $\sum_{n=0}^{\infty} n!x^n$

Question

I need help with how to compute the radius of convergence of the power series $\sum_{n=0}^{\infty} n!x^n$?

I was thinking of using the ratio test but am unsure of how to go about it. Any help would be greatly appreciated, thanks

Answer 1

Ratio Test: suppose $x\neq 0$ because the series clearly converges if $x=0$, we have: $$\lim_{n\to\infty} \left\vert\frac{(n+1)!x^{n+1}}{n!x^n}\right\vert=\lim_{n\to\infty} (n+1)|x|$$ What do you know about the above limit? What does the ratio test then conclude for $x\neq 0$?

Answer 2

The ratio test does apply !

Putting $a_n=n!$ we see that :

$$\left|\frac{a_{n+1}}{a_n}\right|=\frac{(n+1)!}{n!}=n+1\underset{n\to\infty}{\longrightarrow}+\infty$$

Hence : $R=0$

In other words, this power series converges solely for $x=0$.