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Math Help - Maclaurin Series Problem

  1. #1
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    Maclaurin Series Problem

    Hi,

    I'm having trouble finding the Maclaurin series using a known series. I can find the series using the Maclaurin formula but I can't do it using a known series. Does anyone have any idea on which series to use and how I can use it. The series the I have considered was the Binomail series but I cant seem to get it to work for this problem.

    Thanks, DLL.
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  2. #2
    o_O
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    Yep. Write: f(x) = \frac{1}{\sqrt{4-x}} = \left(4-x\right)^{-\frac{1}{2}} = \left[4\left(1 - \tfrac{x}{4}\right)\right]^{-\frac{1}{2}} {\color{white}.} \ \Rightarrow \ f(x) = \tfrac{1}{2} \left(1 - \tfrac{x}{4}\right)^{-\frac{1}{2}}

    From an old post:

    Recall that binomial series are given by: (1+y)^k = 1 + ky + \frac{k(k-1)}{2!}y^2 {\color{white}.} \ +  \ \frac{k(k-1)(k-2)}{3!}y^3 + \cdots + \frac{k(k-1)(k-2)\cdots(k - n+1)}{n!}y^n + \cdots

    for any k \in \mathbb{R} and if |y| < 1. Here, y = (-1)\tfrac{x}{4} and k = -\tfrac{1}{2} .
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