binomial function Taylor series

We have a binomial function $f_m(x)=(1+x{)}^m$ where $m\in \mathbb{R}$.

If $m$ is a positive integer, then the binomial function $f_m$ is a polynomial, and its Taylor series would be the same polynomial.

If $m$ is not a positive integer, then its Taylor series has infinitely many non-zero terms. The Taylor series converges for $|x|<1$ and is given by:

$$T(x)=1+\sum _{n=1}^{\infty }\left(\genfrac{}{}{0ex}{}{m}{n}\right)x^n$$

with binomial coefficients $\left(\genfrac{}{}{0ex}{}{m}{1}\right)=m$, $\left(\genfrac{}{}{0ex}{}{m}{2}\right)=\frac{m(m-1)}{2!}$, and so on.