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Thread: power series representation

  1. #1
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    power series representation

    Hi,

    I was trying to find a power series for function $\displaystyle ln((1+x)/(1-x))$.

    I separated this into $\displaystyle ln(1+x)-ln(1-x)$ and used differentiation & integration to get $\displaystyle \sum_{n=0}^\infty\frac{(-1)^n(x)^{n+1}}{n+1} + \sum_{n=0}^\infty\frac{x^{n+1}}{n+1}$

    The answer says $\displaystyle \sum_{n=0}^\infty\frac{2x^{2n+1}}{2n+1}$. I have no idea how this answer was derived. Can anyone explain to me how to get this answer?

    Thanks!
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    Note if n is even (or 0 ) the terms add and if n is odd they cancel so you have only odd powers and all are positive
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  3. #3
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    Quote Originally Posted by john52302 View Post
    Hi,

    I was trying to find a power series for function $\displaystyle ln((1+x)/(1-x))$.

    I separated this into $\displaystyle ln(1+x)-ln(1-x)$ and used differentiation & integration to get $\displaystyle \sum_{n=0}^\infty\frac{(-1)^n(x)^{n+1}}{n+1} + \sum_{n=0}^\infty\frac{x^{n+1}}{n+1}$

    The answer says $\displaystyle \sum_{n=0}^\infty\frac{2x^{2n+1}}{2n+1}$. I have no idea how this answer was derived. Can anyone explain to me how to get this answer?

    Thanks!
    $\displaystyle \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ...
    $

    $\displaystyle \ln(1-x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - ...$

    $\displaystyle \ln(1+x) - \ln(1-x) = 2\left(x + \frac{x^3}{3} + \frac{x^5}{5} + ... \right)
    $
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