Question

How to apply ratio test on series (e^n)/n^e) to determine of series converges or diverges?

Answer 1

Converges.

Explanation:

If `a_n=e^n/n^e,` we define `a_(n+1)=e^(n+1)/(n+1)^e`

Now, the ratio test tells us that if

`L=lim_(n->oo)|a_(n+1)/a_n|<1,` `a_n` converges. If `L>1, a_n` diverges. So, let's find this limit:

`lim_(n->oo)|e^n/n^e*(n+1)^e/(e^(n+1))|=lim_(n->oo)|(e^n(n+1)^e)/(n^ee^(n)e)|=lim_(n->oo)|((n+1)^e/(en^e))|=1/elim_(n->oo)|(n+1)^e/n^e|=1/elim_(n->oo)|((n+1)/n)^e|=1/e|1^e|=1/e`

So, `L=1/e<1`, and the series converges.