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Convergence of Infinite Series

  • Thread starter aostraff
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  • #1
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I need some help on determining whether this infinite series converges (taken from Spivak for those curious):

[tex]\sum_{n=1}^{\infty } \frac{1}{n^{1+1/n}}[/tex]

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.
 
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Answers and Replies

  • #2
Dick
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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
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Yeah it does. So it converges. Thanks.
 
  • #4
Dick
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Yeah it does. So it converges. Thanks.
No it doesn't! What are you going to compare with? Are you sure it converges???!
 
  • #5
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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
Dick
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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
lanedance
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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)
 
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