Taylor Series with Examples
In this lesson we will learn about taylor series and with some examples of deriving taylor series of functions.
What is Taylor series ?
Taylor series is a representation of function as infinite sum of derivatives at a point.
With the help of taylor series we could write a function as sum of its derivates at a point.
Suppose we have a function f(x) then we can write it as :
In general way taylor series formula can be written as:
$ \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n !}(x-a)^{n} $
Taylor Series Examples
Taylor series of sinx
To find the taylor series of sinx we find the nth derivative of sinx which is :
$ f^{(n)}(x)=\sin \left(x+\frac{n \pi}{2}\right) $
Evaluating the function and its derivatives at x = 0 we obtain
$ \begin{aligned} f(0) &=\sin 0=0 \\ f^{\prime}(0) &=\sin (\pi / 2)=1 \\ f^{\prime \prime}(0) &=\sin \pi=0 \\ f^{\prime \prime \prime}(0) &=\sin (3 \pi / 2)=-1 \end{aligned} $
And So therefore expension of sinx at x=0 which also know as Maclaurin expension is given by :
$ \sin x=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\cdots $
Taylor series of cosx at $ x=\pi / 3 $
As in above example the nth derivative of cosx is given by
$ f^{(n)}(x)=\cos \left(x+\frac{n \pi}{2}\right) $
Evaluating the function and its derivatives at $ x=\pi / 3 $ we obtain
$ \begin{aligned} f(\pi / 3) &=\cos (\pi / 3)=1 / 2 \\ f^{\prime}(\pi / 3) &=\cos (5 \pi / 6)=-\sqrt{3} / 2 \\ f^{\prime \prime}(\pi / 3) &=\cos (4 \pi / 3)=-1 / 2 \end{aligned} $
Thus the taylor expension of cosx at $ x=\pi / 3 $ is given by :
$ \sin x=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\cdots $
$ \cos x=\frac{1}{2}-\frac{\sqrt{3}}{2}(x-\pi / 3)-\frac{1}{2} \frac{(x-\pi / 3)^{2}}{2 !}+\cdots $
Maclaurin series examples
Maclaurin series is a special case of taylor series when taylor series expension done at a=0 .
Maclaurin series examples
Maclaurin series is a special case of taylor series when taylor series expension done at a=0 .
Maclaurin series expension of sinx
$ \sin x=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}+\cdots $ for $ -\infty \lt x \lt -\infty $
Maclaurin series expension of cosx
$ \cos x=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\frac{x^{6}}{6 !}+\cdots $ for $ -\infty \lt x \lt -\infty $
Maclaurin series expension of tanx
$ \tan ^{-1} x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\cdots $ for $ -1 \lt x \lt 1 $
Maclaurin series expension of ex
$ e^{x}=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\cdots $ for $ -\infty \lt x \lt -\infty $