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Math Help - Related to harmonic series

  1. #1
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    Related to harmonic series

    OK so i missed this day in class and i've been trying to understand why a harmonic series is divergent. One of our homework problems is this:

    We have seen that the harmonic series is a divergent series whose terms approach 0. Show that
     <br />
\sum {ln(1+\frac{1}{n}})<br />
    Is another series with this property.

    With infinite on the top and n=1 on the bottom of the sum sign. I don't know how to do that in LaTex.

    My teacher gave use a hint as to how to solve the problem:

    The sum separates into ln(n+1)-ln(n)

    If anyone could show me how to solve this problem, explain harmonic series for me, or direct me to a website that might help i'd really appreciate it. Thanks ^.^
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  2. #2
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    Quote Originally Posted by mortalapeman View Post
    OK so i missed this day in class and i've been trying to understand why a harmonic series is divergent. One of our homework problems is this:

    We have seen that the harmonic series is a divergent series whose terms approach 0. Show that
     <br />
\sum {ln(1+\frac{1}{n}})<br />
    Is another series with this property.

    With infinite on the top and n=1 on the bottom of the sum sign. I don't know how to do that in LaTex.

    My teacher gave use a hint as to how to solve the problem:

    The sum separates into ln(n+1)-ln(n)

    If anyone could show me how to solve this problem, explain harmonic series for me, or direct me to a website that might help i'd really appreciate it. Thanks ^.^
    You'll find the proof here (classic) http://en.wikipedia.org/wiki/Harmoni...s_(mathematics)

    As for the second problem

    \sum_{n = 1}^\infty \ln \left( 1 + \frac{1}{n} \right)= \sum_{n = 1}^\infty \ln (n+1) - \ln n. Stop after N terms and write some of them out

    S_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)

    Everthing cancels except

    S_N = - \ln 1 + \ln(N+1) = \ln(N+1)

    Now as \lim_{N \to \infty} S_N \to \infty so the series diverges.
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