How do you use the Ratio Test on the series #sum_(n=1)^oon^n/(n!)# ?

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Answer 1 · 2014-10-07T09:49:53

Recall: The Definition of Number #e#

#e=lim_{n to infty}(1+1/n)^n#.

(Note: This can be derived suing l'Hopital's Rule as well.)

Now, let us examine the convergence of the posted series.

#lim_{n to infty}|{a_{n+1}}/{a_{n}}|=lim_{n to infty}|(n+1)^{n+1}/{(n+1)!}cdot{n!}/{n^n}|#

by cancelling out common factors,

#=lim_{n to infty}{(n+1)^n}/{n^n}#

by simplifying a bit further,

#=lim_{n to infty}(1+1/n)^n=e ge 1#

Hence, the series diverges.

I hope that this was helpful.