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Homework Help: Infinite series question

  1. Oct 13, 2005 #1
    Hello,
    Here's the question:
    Does the series SUM log(1+1/n) converge or diverge?

    I wrote out the nth partial sums like this:

    log(1+1) + log(1+1/2) + log(1+1/3)+..........+log(1+1/n)

    It looks to me like the limit of the thing inside the parentheses goes to 1 as n goes to infinity, making the limit of the entire thing 0. So I say it converges.

    One of my classmates says that SUM 1/n diverges, so this one does too. I can't disagree with him, but I fail to see the relevance.

    I'm confused. Any clarification will be appreciated.
    CC
     
  2. jcsd
  3. Oct 13, 2005 #2

    shmoe

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    Science Advisor
    Homework Helper

    The terms going to zero is a necessary but not sufficient condition for convergence of the series (1/n is your basic counterexample).

    They probably have the limit comparison test in mind. You can also look at the integral comparison test, you can find an antiderivative of log(1+1/n) easily enough.
     
  4. Oct 13, 2005 #3
    The series diverges ..
    Take the sequance of partial sums ..
    [tex] S_n = \sum_{k=1} ^n\log \left( 1 + \frac 1k \right) = \sum_{k=1} ^n \log \left ( \frac { k+1} { k} \right )
    = \sum_{k=1} ^n \log (k+1) - \log (k) \mbox{ ( Telescoping series } [/tex]

    [tex] = (\log 2 - \log 1) + (\log 3 - \log 2) +....... + (\log n - \log (n-1) ) + (\log (n+1) - \log n) [/tex]

    [tex]= - \log 1 + \log ( n+1 ) = \log (n+1) [/tex]
    [tex] \lim _ { n \rightarrow \infty } S_n = \lim _ { n \rightarrow \infty } \log ( n+1) = \infty
    \Longrightarrow \sum_{n=1} ^ {\infty}\log \left( 1 + \frac 1n \right) \mbox{ diverges } [/tex]
     
    Last edited: Oct 13, 2005
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