Hi Am working through heaps of series problems in which i'm expected to show the series converges or diverges by applying both root and ratio test,

For the series;

$\displaystyle

\sum_{n=1}^{Inf} \frac{2^n}{n!}

$

I went ahead and applied to ratio test getting;

limit as n --> inf of 2/(n+1) = 0 hence 0 < 1 means convergence,

but i then applied the root test and ended up with;

lim as n--> inf of 2/((n!)^(1/n)) which i figured to be = 2/1 hence > 1 and divergent,

Clearly i ahve stuffed one of these up and i'm a bit sketchy with limits so could be there but can someone tell me which i've stuffed up and how?

Many thanks in advance.