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Thread: Maclaurin Series

  1. #1
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    Maclaurin Series

    How would I find the maclaurin series for the following function: $\displaystyle f(x) = \frac{3}{(1-x)^2}$.

    The first, second, and third derivatives are:$\displaystyle \frac{6}{(1-x)^3}, \frac{18}{(1-x)^4}, \frac{72}{(1-x)^5}$.

    $\displaystyle f(x)=\Sigma \frac{f^{(n)}(0)}{n!}x^n = 3 + 6x +\frac{18}{2!}x^2 + \frac{72}{3!}x^3 + ...$.

    Here's where I'm stuck.

    Any help would be greatly appreciated.
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  2. #2
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    Re: Maclaurin Series

    $\displaystyle f(x) = \frac{3}{(1-x)^2}$

    let $\displaystyle F(x)$ be the antiderivative of $\displaystyle f(x)$ ...

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

    $\displaystyle F'(x) = f(x) = \frac{3}{(1-x)^2} = 3(1 + 2x + 3x^2 + ... )$
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  3. #3
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    Re: Maclaurin Series

    Quote Originally Posted by statmajor View Post
    How would I find the maclaurin series for the following function: $\displaystyle f(x) = \frac{3}{(1-x)^2}$.

    The first, second, and third derivatives are:$\displaystyle \frac{6}{(1-x)^3}, \frac{18}{(1-x)^4}, \frac{72}{(1-x)^5}$.

    $\displaystyle f(x)=\Sigma \frac{f^{(n)}(0)}{n!}x^n = 3 + 6x +\frac{18}{2!}x^2 + \frac{72}{3!}x^3 + ...$.

    Here's where I'm stuck.

    Any help would be greatly appreciated.
    The easiest way would be to notice that $\displaystyle \displaystyle 3\,\frac{d}{dx}\left(\frac{1}{1 - x}\right) = \frac{3}{(1 - x)^2}$ and that $\displaystyle \displaystyle \frac{1}{1 - x}$ is the closed form of the geometric series $\displaystyle \displaystyle \sum_{n = 0}^{\infty}x^n$ for $\displaystyle \displaystyle |x| < 1$.
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