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Thread: Power Series Representation

  1. #1
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    Power Series Representation

    Find a power series representation for the function f(x)=ln(1-x^2). Write the series in sigma notation. For which values of x is the power series representation of f valid?
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  2. #2
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    Quote Originally Posted by kiddopop View Post
    Find a power series representation for the function f(x)=ln(1-x^2). Write the series in sigma notation. For which values of x is the power series representation of f valid?
    for $\displaystyle |x|<1$ ...

    $\displaystyle \frac{1}{1-x} = 1 + x + x^2 + x^3 + ...$

    integrate ...

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

    for $\displaystyle x = 0$ , $\displaystyle C = 0$ ...

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

    $\displaystyle \ln(1-x^2) = -\left(x^2 + \frac{x^4}{2} + \frac{x^6}{3} + \frac{x^8}{4} + ... \right)$

    I'll leave you to find the sigma notation for the series.
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  3. #3
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    Power Series For an Indefinite Integral

    Find a power series for the indefinite integral of cos(x^2). Use both sigma notation and expanded form.
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  4. #4
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    Quote Originally Posted by kiddopop View Post
    Find a power series for the indefinite integral of cos(x^2). Use both sigma notation and expanded form.
    start w/ the series for cos(x) ...

    $\displaystyle \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ...
    $

    ... and proceed from there.


    btw, next time ... start a new problem w/ a new thread.
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  5. #5
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    Ok. Thank you. And I thought I had. Oops.
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