Power series expansion involving non integer exponent

Question

I'm working on a real and complex analysis course right now and one power series question has me really stumped:

Let $f(x)=\frac1{(1-x)^{3/2}}$ for every $z\in\mathbb R\setminus\{1\}$.
(a) Calculate the power series expansion of $f$ around $x_0=\pi$.

I'm not sure what to do with the non integer in the exponent, as my initial plan of differentiating the power series of 1/(1-x) won't work.

Any help on this would be great, thanks!

Answer 1

Why don't you just differentiate $f$ as given?

Answer 2

Hint. You may use the generalized binomial theorem $$(1-x)^{\alpha}=\sum_{n=0}^{+\infty}(-1)^n{\alpha \choose n} x^n$$ for $x\in[0;1), \alpha \in\mathbb{R}$ and a change of variable from $x$ near $\pi$ to $x$ near $0$.

Answer 3

I do not see what is the problem after Olivier Oloa's answer (use the generalized binomial theorem).

Doing so, the series expansion you look for is, around $x=a$ $$\frac{1}{(1-x)^{3/2}}=\frac{1}{(1-a)^{3/2}}+\frac{3 (x-a)}{2 (1-a)^{5/2}}+\frac{15 (x-a)^2}{8 (1-a)^{7/2}}+\frac{35 (x-a)^3}{16 (1-a)^{9/2}}+O\left((x-a)^6\right)$$