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Convergence of Infinite Series

  1. Feb 24, 2009 #1
    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.
     
    Last edited: Feb 24, 2009
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  3. Feb 24, 2009 #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?
     
  4. Feb 24, 2009 #3
    Yeah it does. So it converges. Thanks.
     
  5. Feb 24, 2009 #4

    Dick

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    No it doesn't! What are you going to compare with? Are you sure it converges???!
     
  6. Feb 24, 2009 #5
    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.
     
  7. Feb 24, 2009 #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?
     
  8. Feb 24, 2009 #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)
     
    Last edited: Feb 24, 2009
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