Let $a\in \mathbb R$ and $f(x)$ be and infinitely differentiable function on an interval $I$ containing $a$. Then the one-dimensional Taylor series of $f$ around $a$ is given by
Recall that, in real analysis, Taylor's theorem gives an approximation of a $k$-times differentiable function around a given point by a $k$-th order Taylor polynomial.
For example, the best linear approximation for $f(x)$ is $$f(x)\approx f(a)+f′(a)(x−a).$$ This linear approximation fits $f(x)$ with a line through $x=a$ that matches the slope of $f$ at $a$.
For a better approximation we can add other terms in the expansion. For instance, the best quadratic approximation is $$f(x)\approx f(a)+f'(a)(x−a)+\frac12 f''(a)(x−a)^2.$$
The following applet shows the partial sums of the Taylor series for a given function. Drag the slider to show more terms of the series. Drag the point a or change the function.
Suppose that a function $f$ is analytic throughout a disk $|z -z_0|< R$, centred at $z_0$ and with radius $R$. Then $f(z)$ has the power series representation \begin{eqnarray}\label{seriefunction} f(z)=\sum_{n=0}^{\infty} a_n(z-z_0)^n,\quad |z-z_0|<R, \end{eqnarray} where \begin{eqnarray} a_n=\frac{f^{(n)}(z_0)}{n!},\quad n=0,1,2,\ldots \end{eqnarray} That is, series (\ref{seriefunction}) converges to $f(z)$ when $z$ lies in the stated open disk.
Every complex power series (\ref{seriefunction}) has a radius of convergence. Analogous to the concept of an interval of convergence for real power series, a complex power series (\ref{seriefunction}) has a circle of convergence, which is the circle centered at $z_0$ of largest radius $R \gt 0$ for which (\ref{seriefunction}) converges at every point within the circle $|z−z_0|=R$. A power series converges absolutely at all points $z$ within its circle of convergence, that is, for all $z$ satisfying $|z − z_0| \lt R$, and diverges at all points $z$ exterior to the circle, that is, for all $z$ satisfying $|z−z_0| \gt R$. The radius of convergence can be:
The radius of convergence can be calculated using the ratio test of convergence. For example, if:
Use the following applet to explore Taylor series representations and its radius of convergence which depends on the value of $z_0$.
On the left side of the applet below, a phase portrait of a complex function is displayed. On the right side, you can see the approximation of the function through it's Taylor polynomials at the blue base point $z_0$. The complex function, the base point $z_0$, the order of the polynomial (vertical slider) and the zoom (horizontal slider) can be modified.
A Taylor series with centre $z_0=0$ \[ f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}z^n \] is referred to as Maclaurin series.
Some important Maclaurin series are: \begin{eqnarray*} \displaystyle \frac{1}{1-z}&=& \sum_{n=0}^{\infty} z^n, \quad |z|\lt 1; \\ \displaystyle e^z &=& \sum_{n=0}^{\infty} \frac{z^n}{n!}, \quad |z|\lt \infty;\\ \displaystyle \sin z &=& \sum_{n=0}^{\infty} (-1)^n\frac{z^{2n+1}}{(2n+1)!}, \quad |z|\lt \infty;\\ \displaystyle \cos z &=& \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n}}{(2n)!}, \quad |z|\lt \infty;\\ \displaystyle \sinh z &=& \sum_{n=0}^{\infty} \frac{z^{2n+1}}{(2n+1)!}, \quad |z|\lt \infty;\\ \displaystyle \cosh z &=& \sum_{n=0}^{\infty} \frac{z^{2n}}{(2n)!}, \quad |z|\lt \infty; \end{eqnarray*}
Exercise: Find the Maclaurin series expansion of the function \[ f(z)=\frac{z}{z^4+9} \] and calculate the radius of convergence.
Note: The applet was originally written by Aaron Montag using CindyJS. The source can be found at GitHub.
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