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Math Help - Power Series Help

  1. #1
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    Power Series Help

    I have a Question in regards to a Power series questions that I have almost completed.

    My Question looks like this:
    The function f(x)= 5xarctan(4x) Find:
    C1: Which I found to be 0
    C2: Which I found to be 0
    C3: Which I found to be 20
    C4: WHICH I CANT FIGURE OUT, I thought it was 20x^4/3 But it is not right.
    As well Find R: Which I cant figure out...

    If anyone could help that would be great!
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  2. #2
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    \tan^{-1} \mu = \sum_{n=0}^{\infty} \frac{\mu^{2n+1} (-1)^n}{n} for all -1 < \mu \leq 1.

    This means if -1 < 4x \leq 1 \implies -\frac14 < x \leq \frac14 then,
    \tan^{-1} 4x = \sum_{n=0}^{\infty} \frac{4^{2n+1} x^{2n+1} (-1)^n}{n}.

    Thus,
    5x\tan^{-1} 4x = \sum_{n=0}^{\infty} \frac{5\cdot 4^{2n+1} x^{2n+2} (-1)^n}{n}.
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