I wanted to know if we can have power series for functions like $x^{\alpha}$, where $\alpha \in \mathbb{R}$. One case we know is that $\alpha \in \mathbb{N} \cup \left\lbrace 0 \right\rbrace$, where $x^{\alpha}$ is already in the power series representation with all but one coefficients zero.
What about other values of $\alpha$? Can we still have a power series (NOT Taylor's series) for $\alpha \in \mathbb{R} \setminus \left( \mathbb{N} \cup \left\lbrace 0 \right\rbrace \right)$? If so, how do we find it?
You certainly can have Puiseux series, which are formal power series with the powers being rational instead of integers, like $$ \sqrt{z(2+z)}=\sqrt{2}\sum_{j=0}^\infty \binom{1/2}{j}2^{-j}z^{(2j+1)/2}. $$ We use them in, e.g., parametrising a curve near a branch point.