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Determine the convergence/divergence of this series with all possible tests

  1. Jul 23, 2012 #1
    1. The problem statement, all variables and given/known data
    series from n=0 to infinity of
    (n+1)/3^n

    Did I do this correctly? Also, I'm pretty positive that I can use the comparison test/limit comparison test, but I can't seem to figure out what to compare it to, could someone shed a little light? For the integral test, the integral is quite intimidating and I don't really know where to start, could someone suggest what integration technique to try?

    relevant equations:
    Ratio Test


    3. The attempt at a solution
    so first I used the ratio test
    lim n->infinity |((n+2)/3^(n+1)) * ((3^n)/(n+1))|
    = lim n->infinity (n+2)/(3(n+1))
    dividing by n/n
    you have the limit = 1/3<1, therefore it converges by the ratio test.
     
  2. jcsd
  3. Jul 23, 2012 #2

    micromass

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    That's good.

    If you want to do the comparison test, then maybe apply

    [tex](n+1)<2^n[/tex]

    for large n?
     
  4. Jul 23, 2012 #3
    So then I would have a convergent geometric series since 2/3<1.
    Wow, beautiful. I would've never thought of that on my own, been staring at it for 30minutes trying to compare it to random things.
     
  5. Jul 24, 2012 #4
    You could simply divide and you would have two geometric series.

    Also trivially, it's probably 1 to infinity, because of the n^3 on the bottom.
     
  6. Jul 24, 2012 #5
    I'm confused on where the n^3 came from. Was that a typo and you meant 3^n?
     
  7. Jul 24, 2012 #6

    Mark44

    Staff: Mentor

    I think that johnqwertyful misread your post.
     
  8. Jul 24, 2012 #7
    Oops, misread.
     
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