Taylor Series 

The Taylor series is a mathematical method used to represent a function as an infinite sum of terms based on its derivatives at a single point. It provides polynomial approximations of complex functions, making them easier to analyze and compute. Commonly used in calculus, physics, and engineering, Taylor series help estimate values of functions like sin x, cos x, and eˣ. When expanded around zero, it becomes a Maclaurin series, a special case of the Taylor series

1.0What is the Taylor Series?

In calculus, the Taylor series of a function is an infinite sum of terms calculated from the values of its derivatives at a single point. It's an essential method for function approximation, numerical analysis, and solving differential equations.

2.0Taylor Series Formula

The Taylor series formula for a function f(x)f(x) expanded about a point aa is:

$f(x)=f(a)+f^{\prime }(a)(x-a)+\frac{f^{\prime \prime }(a)}{2!}(x-a{)}^2+\frac{f^{\prime \prime \prime }(a)}{3!}(x-a{)}^3+\dots$

Or in sigma notation:

$f(x)={\sum }_{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n$

When a = 0, it's called a Maclaurin series (a special case of the Taylor series).

3.0Taylor Series Expansion of Standard Functions

sin x Taylor Series

$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\dots ={\sum }_{n=0}^{\infty }(-1{)}^n\frac{x^{2n+1}}{(2n+1)!}$

cos x Taylor Series

$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\dots ={\sum }_{n=0}^{\infty }(-1{)}^n\frac{x^{2n}}{(2n)!}$

tan x Taylor Series (around 0)

The tan x Taylor series is more complex and only valid in the interval $-\frac{\pi }{2}<x<\frac{\pi }{2}$:

$\tan x=x+\frac{x^3}{3}+\frac{2x^5}{15}+\frac{17x^7}{315}+\dots$

4.0Taylor Series for Two Variables

The Taylor series for two variables extends the single-variable Taylor expansion to functions of two variables f(x, y). It approximates the function around a point (a, b).

Taylor Series Formula for Two Variables:

$\begin{array}{rl}& f(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)+\frac{1}{2!}[f_{xx}(a,b)(x-a{)}^2+\\ & 2f_{xy}(a,b)(x-a)(y-b)+f_{yy}(a,b)(y-b{)}^2]+\dots \end{array}$

Where:

• $f_x,f_y$ : first partial derivatives
• $f_{xx},f_{xy},f_{yy}$ : second partial derivatives

5.0Applications of Taylor Series

• Approximating non-polynomial functions like exponential, logarithmic, trigonometric functions
• Solving differential equations
• Physics and engineering problems involving small-angle approximations
• Computer algorithms (e.g., calculating sine, cosine using series expansion)

Here are 5 solved examples and 5 practice questions on the Taylor Series, including single-variable and two-variable cases:

6.0Solved Examples on Taylor Series

Example 1: Taylor Series of $e^x$ at x = 0

Solution: 

We know:

$f(x)=e^x,f^{(n)}(0)=1\text{ for all }n$

So,

$e^x={\sum }_{n=0}^{\infty }\frac{x^n}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots$

Example 2: Taylor Series Expansion of \ln(1 + x) at x = 0

Solution: 

We know:

$f(x)=\ln (1+x)$

Derivatives:

$\begin{array}{rl}& f^{\prime }(x)=\frac{1}{1+x}\\ & f^{\prime \prime }(x)=-\frac{1}{(1+x{)}^2}\\ & f^{(3)}(x)=\frac{2}{(1+x{)}^3},\text{ etc. }\end{array}$

At x = 0:

$\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\dots ={\sum }_{n=1}^{\infty }(-1{)}^{n+1}\frac{x^n}{n}$

Example 3: Maclaurin Series of \sin x

Solution: 

We have:

So,

$f(x)=\sin x,f^{(n)}(0)=0,1,0,-1,\dots$

Example 4: Taylor Series Expansion of $f(x,y)=e^{x+y}at(0,0)$

Solution: 

All derivatives of $e^{x+y}are{e}^{x+y}$ . At (0, 0), each derivative = 1.

$f(x,y)=1+(x+y)+\frac{(x+y{)}^2}{2!}+\frac{(x+y{)}^3}{3!}+\dots =e^{x+y}$

Example 5: Taylor Series of f(x) = \cos x around x = 0

Solution: 

Derivatives:

$\begin{array}{rl}& f(x)=\cos x\\ & f^{\prime }(x)=-\sin x\\ & f^{\prime \prime }(x)=-\cos x\\ & f^{(3)}(x)=\sin x\end{array}$

So,

$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\dots ={\sum }_{n=0}^{\infty }(-1{)}^n\frac{x^{2n}}{(2n)!}$

Example 6:  What is the Taylor series of $e^x$?

$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots$

7.0Practice Questions on Taylor Series

Question 1: Find the Taylor series expansion of $f(x)={\tan }^{-1}$ x around x = 0.

Question 2: Find the Taylor series expansion of $f(x,y)=\ln (1+x+y)$ at (0, 0) up to the second-degree term.

Question 3: Expand $f(x)=\frac{1}{1-x}$ using Taylor series around x = 0. Write the first 4 terms.

Question 4: Use Taylor series to approximate $\sqrt{1+x}$ up to 3 terms.

Question 5: Find the Taylor series of $f(x)=\sin \left(x^2\right)$ around x = 0.