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### Taylor Series

#### Definition

A Taylor series is a power series expansion of a function $f(x)$ about a given point. The Taylor series expansion of a differentiable function $f(x)$ about a point $x=c$ is given by:

$\sum_{n=0}^{\infty}\dfrac{f^{(n)}(c)}{n!}(x-c)^n=f(c)+f'(c)(x-c)+\dfrac{f''(c)}{2!}(x-c)^2+\dfrac{f'''(c)}{3!}(x-c)^3+\ldots$

The Taylor series expansion about the point $c=0$ is known as a Maclaurin series expansion.

Taylor series expansions have many applications, including evaluating definite integrals, finding the limit of a function and approximating the value of an expression.

#### Worked Example

###### Example 1

Use the first two terms of a Taylor expansion to approximate $\cos{\left(\dfrac{4\pi}{5}\right)}$.

###### Solution

Here the function to expand is $f(x)=\cos{x}$. Recall that the first two terms of a Taylor expansion about the point $x=c$ are given by $f(x)\approx f(c)+f'(c)(x-c).$

The derivative of $\cos{x}$ with respect to $x$ is $-\sin{x}$, so the first two terms of the Taylor series expansion for $\cos{x}$ are: $f(x)\approx \cos{c}-(x-c)\sin{c}.$

To approximate $\cos{\left(\dfrac{4\pi}{5}\right)}$, a suitable choice for $c$ must be made.

First note that: $\frac{4\pi}{5}=\frac{16\pi}{20}=\frac{(15+1)\pi}{20}=\frac{3\pi}{4}+\frac{\pi}{20}$.

Since $\cos{\left(\dfrac{3\pi}{4}\right)}$ and $\sin{\left(\dfrac{3\pi}{4}\right)}$ are commonly known trigonometric ratios, choose $c=\dfrac{3\pi}{4}$. Substituting $x=\dfrac{4\pi}{5}$ and $c=\dfrac{3\pi}{4}$ into the expansion gives

\begin{align} \cos{\left(\frac{8\pi}{10}\right)} &\approx \cos{\left(\frac{3\pi}{4}\right)}-\left(\frac{8\pi}{10}-\frac{3\pi}{4}\right)\sin{\left(\frac{3\pi}{4}\right)} \\ &= -\frac{1}{\sqrt{2} }-\frac{\pi}{20}\cdot\frac{1}{\sqrt{2} } \\ &= -\frac{1}{\sqrt{2} }\left(1+\frac{\pi}{20}\right). \end{align}

#### Video Examples

###### Taylor Expansion

Prof. Robin Johnson finds the Taylor expansion of $(6+4x)^{\large{^1/_3}}$ about $x=-1$.

##### Applications of Taylor Series
###### Example 1

Prof. Robin Johnson uses Taylor series to approximate $\sqrt{1.1}$.

###### Example 2

Prof. Robin Johnson uses Taylor series to approximate $^3\sqrt{26}$.

###### Example 3

Prof. Robin Johnson uses Taylor series to approximate $\ln{(0.95)}$.

###### Example 4

Prof. Robin Johnson uses Taylor series to approximate $\sin{\left(\dfrac{3\pi}{10}\right)}$.

#### Workbooks

These workbooks produced by HELM are good revision aids, containing key points for revision and many worked examples.