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Thread: Please Help With Summations!

  1. #1
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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; Apr 30th 2008 at 06:44 AM.
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  2. #2
    Super Member wingless's Avatar
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    $\displaystyle \sum_{k=1}^{\infty} \ln \frac{k+1}{k}$

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

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

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


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


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

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

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

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

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

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