How to choose the Bn for limit comparison test?

If the An is (e^(1/n))/n
how would you determine what bn to use to compare with this?

1 Answer
Dec 22, 2017

Note that e^{1/n}>1 for all integers n>0. Therefore, we expect that sum_{n=1}^{infty}e^{1/n}/n will diverge. Try comparing it to the divergent harmonic series sum_{n=1}^{infty}1/n to show this with the limit comparison test (so use b_{n}=1/n).

Explanation:

Let a_{n}=e^{1/n}/n and b_{n}=1/n, noting that a_{n} > b_{n} > 0 for all integers n>0.

Now compute lim_{n->infty}a_{n}/b_{n}. We are"hoping" it is a positive number and not infty, which will allow us to say that sum_{n=1}^{infty}e^{1/n}/n diverges by the Limit Comparison Test since we know that the harmonic series sum_{n=1}^{infty}1/n diverges.

But clearly, lim_{n->infty}a_{n}/b_{n}=lim_{n->infty}e^{1/n}=1, a positive number (and not infty). We are done.