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Math Help - infinite series problem with log

  1. #1
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    Question infinite series problem with log


    This problem is giving me a hard time, i have no clue how to solve for it to be able to state if its convergent or divergent. The problem is:
    signma : log(n+1) - log(n) and the limits are n=1 to infinity

    thanks in advance for the help
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  2. #2
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    Quote Originally Posted by [s]arah View Post

    This problem is giving me a hard time, i have no clue how to solve for it to be able to state if its convergent or divergent. The problem is:
    signma : log(n+1) - log(n) and the limits are n=1 to infinity

    thanks in advance for the help
     <br />
\sum_{i=1}^\infty \ln (n+1) - \ln n<br />

    Stop the series at N and write out some of the terms

     <br />
S_N=\sum_{i=1}^N\ln (n+1) - \ln n = \left(\ln 2 - \ln 1\right) + \left(\ln 3 - \ln 2\right) + \left(\ln 4 - \ln 3\right) + \cdots + \left(\ln N - \ln (N-1)\right) + \left(\ln (N+1) - \ln N\right)

    then cancel terms giving

     <br />
S_N = \sum_{i=1}^N\ln (n+1) - \ln n = \ln (N+1)

    Then \lim_{N \to \infty } S_N
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