## Convergence and divergence of Taylor expansions

The following Taylor expansions of functions of one-variable $$x \in \mathbb{R}$$ are well-known: \begin{align*} (1+x)^\alpha & = \sum_{n=0}^\infty \begin{pmatrix}\alpha \\ n\end{pmatrix} x^n, \\ \begin{pmatrix}\alpha \\ n\end{pmatrix} & = \begin{cases} \dfrac{\alpha\cdot(\alpha-1)\cdot \dotsb \cdot(\alpha-n+1) }{n!} &\ (n=1,2,3,\dotsc), \\ 1 &\ (n=0), \end{cases} \quad \alpha\in\mathbb{C}, \end{align*} If $$\alpha=0,1,2,\dotsc$$, then $$(1+x)^\alpha$$ is a polynomial of order $$\alpha$$, $\begin{pmatrix}\alpha \\ n\end{pmatrix} = \begin{cases} \dfrac{\alpha!}{(\alpha-1)!n!} &\ (n=0,\dotsc,\alpha), \\ 0 &\ (n>\alpha), \end{cases}$ and the above formula is the binomial theorem. So we consider the case of $$\alpha\ne0,1,2,\dotsc$$ below. The power series converges uniformly in any closed interval $$[a,b]$$ contained in the open interval $$(-1,1)$$, and diverses outside the closed interval $$[-1.1]$$. The behavior at $$x=\pm1$$ of the series depends on $$\alpha$$. Some series converge and another series diverge. A MATLAB code taylorpower.m. creates the animation of the convergence and the divergence of the Taylor series of the following functions: \begin{align*} (1+x)^{-1} & = \sum_{n=0}^\infty (-1)^{n} x^n, \\ (1+x)^{-1/2} & = \sum_{n=0}^\infty \frac{(-1)^{n}(2n)!}{2^{2n}(n!)^2} x^n, \\ \log(1+x) & = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} x^n, \\ (1+x)^{1/2} & = 1 + \sum_{n=1}^\infty \frac{(-1)^{n-1}(2n-2)!}{2^{2n-1}(n-1)!n!} x^n \end{align*}