Find the value of $x >0$ for which the series
$1 + \frac{x}{1!} + \frac {2^2x^2} {2!} + \frac {3^3x^3}{ 3!} +....+ \frac {n^nx^n} {n!}$+......................... converges
My solution : By D - Alembert 's Ratio Test ,the series will converges if $x < \frac {1}{e} $ and diverges if $x> \frac {1}{e}$
Is Its true/false ??..Pliz check my answer
thanks in advance
$$1 + \frac{x}{1!} + \frac {2^2x^2} {2!} + \frac {3^3x^3}{ 3!} +....+ \frac {n^nx^n} {n!}$$
Root test gives $$ \frac {nx}{n!^{1/n}} \to ex $$
Thus it converges for $|x|<1/e$