# Taylor Series

## For Real Functions

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

$f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+\cdots$
which can be written in the most compact form:
$f(x)=\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n.$

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.

Sorry, the applet is not supported for small screens. Rotate your device to landscape. Or resize your window so it's more wide than tall.

## For Complex Functions

Suppose that a function $f$ is analytic throughout a disk $|z -z_0|< R_0$, centred at $z_0$ and with radius $R_0$. 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_0), \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.

## Dynamic Exploration

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 and the zoom can be modified.

f(z) =

You can also select one function of the following list:

Sorry, the applet is not supported for small screens. Rotate your device to landscape. Or resize your window so it's more wide than tall.

Note: The applet was originally written by Aaron Montag using CindyJS. The source can be found at GitHub.

NEXT: The Mandelbrot Set