Convergence of Infinite Series

[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.

 

[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?

 

[aostraff]

Yeah it does. So it converges. Thanks.

 

[Dick]

Yeah it does. So it converges. Thanks.

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

 

[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.

 

[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?

 

[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)