Homework Help: Convergence of Infinite Series

1. Feb 24, 2009

aostraff

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: Feb 24, 2009
2. Feb 24, 2009

Dick

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?

3. Feb 24, 2009

aostraff

Yeah it does. So it converges. Thanks.

4. Feb 24, 2009

Dick

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

5. Feb 24, 2009

aostraff

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.

6. Feb 24, 2009

Dick

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?

7. Feb 24, 2009

lanedance

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: Feb 24, 2009