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Math Help - Log(x+1)

  1. #1
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    Log(x+1)

    \int^x_0\sum^\infty_{n=1}\frac{(-1)^{n-1}t^{n}}{n}=\int^x_0\sum^\infty_{n=1}(-1)^{n-1}t^{n-1}dt\underbrace{=}_{\mbox{justify}}\int^x_0\frac{1  }{1+t}dt=\log(1+x)

    Ok so I need to prove that the middle equality it true, but I don't have the slightest clue where to begin.
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Log(x+1)

    You have to prove that:
    \sum_{n=1}^{\infty}(-1)^{n-1}t^{n-1}=\frac{1}{1+t}
    It can be useful to write a few terms of the summation:
    \sum_{n=1}^{\infty}(-1)^{n-1}t^{n-1}=1-t+t^2-t^3+...+(-1)^{k-1}t^{k-1}+...

    Do you notice a 'special' serie? Can you continue?
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  3. #3
    Member sbhatnagar's Avatar
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    Re: Log(x+1)

    1-t+t^2-t^3+t^4-.... is an infinite geometric series with a=1 and r=-t.

    Therefore, 1-t+t^2-t^3+t^4-.... = \frac{a}{1-r}
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