# Taylor Series

A function $f$ is said to be continuous at a point $x=a$ , if the limit of $f\left(x\right)$ as $x$ approaches $a$ is equal to the value of $f\left(c\right)$ .The function is said to be continuous if this is true for every value of $a$ in the domain. Polynomial functions are continuous.

A function is said to be differentiable at a point if the slope of the tangent line of the points from the left is approaching the same value as the slope of the tangent of the points from the right. This value is called the derivative at this point.

The foremost condition for a function to be differentiable is that the function should be continuous.

We can find the derivatives of higher order by repeating the process and they are denoted by ${f}^{\prime },{f}^{″},{f}^{‴}$ etc.

Suppose that $f\left(x\right)$ is a function, and that all the derivatives ${f}^{\prime },{f}^{″},{f}^{‴},$ , etc. exist at $x=a$ . Then the Taylor Series of $f\left(x\right)$ is the power series

$f\left(a\right)+\frac{{f}^{\prime }\left(a\right)}{1!}\left(x-a\right)+\frac{{f}^{″}\left(a\right)}{2!}{\left(x-a\right)}^{2}+\frac{{f}^{‴}\left(a\right)}{3!}{\left(x-a\right)}^{3}+\cdots$

or, in sigma notation,

$\underset{n=0}{\overset{\infty }{\sum }}\frac{{f}^{\left(n\right)}\left(a\right)}{n!}{\left(x-a\right)}^{n}$

A Maclaurin series is a Taylor series in the case where $a=0$ .

The partial sums of the Taylor series are called Taylor polynomials . These can be used to approximate the function in the neighborhood of $x=a$ .

Example:

Find the Taylor polynomials for the function

$f\left(x\right)=\mathrm{sin}\left(x\right)$

around $x=0$ .

$\begin{array}{l}{T}_{0}\left(x\right)=f\left(0\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\mathrm{sin}\left(0\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=0\end{array}$

$\begin{array}{l}{T}_{1}\left(x\right)=f\left(0\right)+{f}^{\prime }\left(0\right)\left(x-0\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\mathrm{sin}\left(0\right)+\left(\mathrm{cos}\left(0\right)\right)\left(x\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=0+1\cdot x\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=x\end{array}$

$\begin{array}{l}{T}_{2}\left(x\right)=f\left(0\right)+{f}^{\prime }\left(0\right)\left(x-0\right)+\frac{{f}^{″}\left(0\right){\left(x-0\right)}^{2}}{2!}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\mathrm{sin}\left(0\right)+\left(\mathrm{cos}\left(0\right)\right)\left(x\right)+\frac{\left(-\mathrm{sin}\left(0\right)\right){x}^{2}}{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=x\end{array}$

$\begin{array}{l}{T}_{3}\left(x\right)=f\left(0\right)+{f}^{\prime }\left(0\right)\left(x-0\right)+\frac{{f}^{″}\left(0\right){\left(x-0\right)}^{2}}{2!}+\frac{{f}^{‴}\left(0\right){\left(x-0\right)}^{3}}{3!}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\mathrm{sin}\left(0\right)+\left(\mathrm{cos}\left(0\right)\right)\left(x\right)+\frac{\left(-\mathrm{sin}\left(0\right)\right){x}^{2}}{2}+\frac{\left(-\mathrm{cos}\left(0\right)\right){x}^{3}}{6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=x-\frac{{x}^{3}}{6}\end{array}$