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Math Help - Please Help With Summations!

  1. #1
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    Apr 2008
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    HURRY- Help With Summations PLEASE!

    I'm studying for a calculus test I have later today and I'm having lots of trouble with these examples. Help with any or all of these is greatly appreciated.

    1. Show that the SUMMATION from k=1 to infinity of ln[(k+1)/k)] diverges although ln[(k+1)/k] --> 0.

    2. Determine whether the series converges or diverges: SUMMATION of 1/(1+2+...+k)

    3. Determine whether the series converges or diverges: SUMMATION of ln(k)/e^k

    4. Determine whether the series converges or diverges: SUMMATION of [k^(k/2)]/k!
    Last edited by seth55; April 30th 2008 at 06:44 AM.
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  2. #2
    Super Member wingless's Avatar
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    \sum_{k=1}^{\infty} \ln \frac{k+1}{k}

    \sum_{k=1}^{\infty} \ln (k+1) - \ln{k}

    \not\ln \not 2 ~- ~\ln 1
    \not\ln \not 3~ - ~\not\ln \not 2
    \not\ln \not 4 ~-~ \not\ln \not 3
    \not\ln \not 5 ~-~ \not\ln \not 4
    ............
    \ln \infty ~-~ \not{...}

    It makes,
    \ln \infty - \ln 1 = \infty


    ---------------
    ---------------


    \sum_{k=1}^{\infty} \frac{1}{1 + 2+3+4+...+k}

    \sum_{k=1}^{\infty} \frac{1}{\frac{k(k+1)}{2}}

    \sum_{k=1}^{\infty} \frac{2}{k(k+1)}

    \sum_{k=1}^{\infty} \frac{2}{k} - \frac{2}{k+1}

    \frac{2}{1} - \not\frac{2}{2}
    \not\frac{2}{2} - \not\frac{2}{3}
    \not\frac{2}{3} - \not\frac{2}{4}
    \not\frac{2}{4} - \not\frac{2}{5}
    ............
    \not\frac{1}{\infty} - \frac{1}{\infty}

    It makes,
    \frac{2}{1} - \frac{1}{\infty} = 2
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