Question

Is the series convergent or divergent using comparison test:

I'll type it below.

Answer 1

\[\sum_{n = 1}^{\infty}e ^{1/n}/n\]

Answer 2

Limit comparison or direct?

Answer 3

I cant think of any thing to compare it to directly.

Answer 4

Should I compare it to 1/n ?

Answer 5

Well, you know that your series is always bigger than 1/n just slightly. And 1/n is the divergent harmonic so I believe thats all you need. Is that:
\[\sum_{n=1}^{\infty}\frac{e^{\frac{1}{n}}}{n}>\sum_{n=1}^{\infty}\frac{1}{n}\]
And 1/n diverges so your series diverges by direct comparison.

Answer 6

Ha, it is really that simple. Thank you!

Answer 7

No problem :P