[Math] 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:

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

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 $0$.