# Convergence of Infinite Series

I need some help on determining whether this infinite series converges (taken from Spivak for those curious):

$$\sum_{n=1}^{\infty } \frac{1}{n^{1+1/n}}$$

I would think the integral test would be most appropriate but it doesn't seem to work (because the integral seems hard). The obvious comparison tests are inconclusive and a ratio test seems inconclusive as well. I'm guessing the best idea right now would be to think of comparisons. Thanks.

Last edited:

Dick
Homework Helper
Your denominator is n^(1+1/n)=n*n^(1/n). Can you figure out the limit of n^(1/n)? Does that suggest a comparison?

Yeah it does. So it converges. Thanks.

Dick
Homework Helper
Yeah it does. So it converges. Thanks.

No it doesn't! What are you going to compare with? Are you sure it converges???!

Uh oh. I'm back to square one. I thought I had it but I was being reckless. I'm not quite sure how the existence of a limit for n^(1/n) helps me.

Dick
Homework Helper
What IS the limit? The existence of a limit implies n^(1/n) has a maximum value. Can you find an M such that n^(1/n)<M. Now what series to compare with?

lanedance
Homework Helper
so n^(1/n) in limit goes to 1....

think about what type of sum this leaves you with as we go towards the limit

it is also< 2 for all n in the sum (actually <=3^(1/3) can show x^(1/x) decreases monotonically for lnx>1)

Last edited: