Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search


Informal Derivation of Taylor Series

We have a function $ f(x)$ and we want to approximate it using an $ n$ th-order polynomial:

$\displaystyle f(x) = f_0 + f_1 x + f_2 x^2 + \cdots + f_n x^n + R_{n+1}(x)
$

where $ R_{n+1}(x)$ , the approximation error, is called the remainder term. We may assume $ x$ and $ f(x)$ are real, but the following derivation generalizes unchanged to the complex case.

Our problem is to find fixed constants $ \{f_i\}_{i=0}^{n}$ so as to obtain the best approximation possible. Let's proceed optimistically as though the approximation will be perfect, and assume $ R_{n+1}(x)=0$ for all $ x$ ( $ R_{n+1}(x)\equiv0$ ), given the right values of $ f_i$ . Then at $ x=0$ we must have

$\displaystyle f(0) = f_0
$

That's one constant down and $ n-1$ to go! Now let's look at the first derivative of $ f(x)$ with respect to $ x$ , again assuming that $ R_{n+1}(x)\equiv0$ :

$\displaystyle f^\prime(x) = 0 + f_1 + 2 f_2 x + 3 f_2 x^2 + \cdots + n f_n x^{n-1}
$

Evaluating this at $ x=0$ gives

$\displaystyle f^\prime(0) = f_1.
$

In the same way, we find

\begin{eqnarray*}
f^{\prime\prime}(0) &=& 2 \cdot f_2 \\
f^{\prime\prime\prime}(0) &=& 3\cdot 2 \cdot f_3 \\
& \cdots & \\
f^{(n)}(0) &=& n! \cdot f_n
\end{eqnarray*}

where $ f^{(n)}(0)$ denotes the $ n$ th derivative of $ f(x)$ with respect to $ x$ , evaluated at $ x=0$ . Solving the above relations for the desired constants yields

\begin{eqnarray*}
f_0 &=& f(0) \\
f_1 &=& \frac{f^{\prime}(0)}{1} \\
f_2 &=& \frac{f^{\prime\prime}(0)}{2\cdot 1} \\
f_3 &=& \frac{f^{\prime\prime\prime}(0)}{3\cdot 2\cdot 1} \\
& \cdots & \\
f_n &=& \frac{f^{(n)}(0)}{n!}.
\end{eqnarray*}

Thus, defining $ 0!\isdef 1$ (as it always is), we have derived the following polynomial approximation:

$\displaystyle \zbox {f(x) \approx \sum_{k=0}^n \frac{f^{(k)}(0)}{k!} x^k}
$

This is the $ n$ th-order Taylor series expansion of $ f(x)$ about the point $ x=0$ . Its derivation was quite simple. The hard part is showing that the approximation error (remainder term $ R_{n+1}(x)$ ) is small over a wide interval of $ x$ values. Another ``math job'' is to determine the conditions under which the approximation error approaches zero for all $ x$ as the order $ n$ goes to infinity. The main point to note here is that the Taylor series itself is simple to derive.


Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA