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

power-series

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


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 $x\neq 0$?

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