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Question:

Compute the Taylor Series of ##\frac{q}{\sqrt{1+x}}## about x = 0.

Attempt at Solution:

Term-wise, I have gotten...

##f(0)+f'(0)+f''(0)+... = 1+1\left(-\frac{1}{2}\right)x+1\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)x^2+1\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)\left(-\frac{5}{2}\right)x^3+...##

I have gotten this to reduce to...

##\sum\limits_{k=0}^{\infty}x^k\left(-\frac{1}{2}\right)^k\frac{(2k-1)!!}{2^k}##

There is definitely a better way to do this. I am not thinking clearly. Additionally, I am not that confident in my answer, given the time of night.