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*}