The Taylor series is a mathematical series representation of a function that can be used to approximate the value of the function at a specific point or interval. The Taylor series expands a function as an infinite sum of terms, where each term in the series is determined by the derivatives of the function evaluated at a specific point.

In the 18^{th} Century, the Taylor series was first introduced by Brook Taylor. Taylor series is used in many fields of mathematics, physics, engineering, numerical analysis, and optimization.

The Taylor series can be used to approximate the value of a function at a specific point, by evaluating a finite number of terms in the series. The more terms that are included in the series, the closer the approximation will be to the actual value of the function.

In this article, we will discuss the definition of the Taylor series, its application in different fields, the Taylor theorem, and perform different examples of the Taylor series.

## Taylor Series Definition:

Taylor Series is the polynomial or the dependent function of an infinite sum of the terms or series of functions. The Taylor series representation of a function f(x) about a point can be written as:

f(x) = f(a) + f'(a)(x-a)/1! + f”(a)(x-a)^{2}/2! + f”'(a)(x-a)^{3}/3! + …

where, f'(a), f”(a), f”'(a), and so on, represent the first, second, third, and higher order derivatives of f(x) evaluated at x=a, respectively. The notation “n!” represents the factorial of “n”, which is the product of all whole numbers from “1 to n”.

The Taylor Series can be written in the form of sigma notation as follow:

f(x) = ∑_{n=1}^{ꝏ} (f^{n}(a)/n!) (x-a)^{n}

Where n!” is the factorial of n (n is the positive integer) and f^{n}(a) is the n^{th}-derivative of function “f” at “x = a”.

## What is Taylor’s Theorem?

Taylor’s theorem is a mathematical theorem that provides a formula for the remainder term in the Taylor series of a function at a particular point. The English mathematician Brook Taylor, who promoted the Taylor series concept in the 18th century, is recognized for having this theorem retain his name.

The Taylor theorem states that any sufficiently smooth function f(x) can be approximated by its nth-degree Taylor polynomial at a particular point “a”, with the remainder term R_{n}(x) given as:

**f(x) = f(a) + f'(a)(x-a)/1! + f”(a)(x-a)**^{2}**/2! + … + f**^{n}**(a)(x-a)**^{n}**/n! + R**_{n}**(x)**

Where R_{n}(x) is the remainder term, which is equal to the n^{th}-derivative of f(x) evaluated at some point c between “a” and “x”, multiplied by the (x-c)^{(n+1)} divided by (n+1)!: which is

**R**_{n}**(x) = [f**^{(n+1)}**(c)(x-c)**^{(n+1)}**] / (n+1)!**

The notation f^{(n)}(x) represents the n^{th}-derivative of f(x) with respect to x.

## Application of Taylor Series:

The Taylor series has numerous applications in mathematics, physics, and other fields, which are given as:

**Numerical Analysis:**The Taylor series can be used to approximate the value of a function at a specific point or interval, which is useful in numerical analysis for solving differential equations and mathematical models.**Calculus:**In calculus, the Taylor series can be used to find the derivatives of a function at a point, as the coefficients of the Taylor series are the derivatives of the function evaluated at that point.**Physics:**The Taylor series is used in physics to approximate the behavior of physical systems, such as the motion of particles, the behavior of waves, and the solution of differential equations that describe physical phenomena.**Computer Science:**The Taylor series is used in computer science to develop numerical algorithms for solving mathematical problems, such as root-finding and integration.

## Examples Of Taylor Series

In this section, we solve the examples of the Taylor series using different functions.

**Example 1:**

Find the Taylor series of “sin(u)” at “a = 1”.

**Solution:**

Step 1: write the given value is equal to function “f(u)”.

f(u) = sin(u)

Step 2: write the formula of the Taylor series.

**f(u) = ∑**_{n=1}^{ꝏ}** (f**^{n}**(a)/n!) (u-a)**^{n}

f(u) = f(a) + f'(a)(u-a)/1! + f”(a)(u-a)^{2}/2! + f”'(a)(u-a)^{3}/3! + …

Step 3: Find the derivatives of the given function.

f’(u) = cos(u) f’’(u) = -sin(u)

f’’’(u) = -cos(u) f’’’’(u) = sin(u)

Step 4: put the given point in the above functions.

f’(1) = cos(1) f’’(1) = -sin(1) f(1) = sin(1)

f’’’(1) = -cos(1) f’’’’(1) = sin(1)

Step 5: Put the above value carefully in the Taylor Series formula written in step 2.

Where “a = 1”.

f(u) = f(a) + f'(a)(u-a)/1! + f”(a)(u-a)^{2}/2! + f”'(a) (u-a)^{3}/3! + …

f(u) = f(1) + f'(1)(u-1)/1! + f”(1)(u-1)^{2}/2! + f”'(1) (u-1)^{3}/3! + …

f(u) = sin(1) + cos(1) (u-1)/1! + [-sin(1)](u-1)^{2}/2! + [-cos(1)] (u-1)^{3}/3! + …

f(u) = sin(1) + cos(1) (u-1)/1! – sin(1) (u-1)^{2}/2! – cos(1) (u-1)^{3}/3! + …

step 6: Find the value of Sin(1) & Cos(1) by trigonometric calculator and put in the above expression.

Sin(1) = 0.84147, Cos(1) = 0.54030

f(u) = (0.84147)+ (0.54030) (u-1)/1! – (0.84147) (u-1)^{2}/2! – (0.54030) (u-1)^{3}/3! + …

**f(u) = (0.84147)+ (0.54030) (u-1)/1! – (0.84147) (u-1)**^{2}**/2! – (0.54030) (u-1)**^{3}**/3! + …**

is the Taylor series of “sin(u)” at “a = 1”.

To get rid of the above lengthy calculations to find the Taylor series of a function, you can try a Taylor polynomial calculator by Allmath (https://www.allmath.com/taylor-series-calculator.php)

**Example 2:**

Find the Taylor series of exponential functions dependent on “u” and point “a = 0”.

**Solution:**

Step 1: write the exponential is equal to function “f(u)”.

f(u) = e^{u}

Step 2: write the formula of the Taylor series.

**f(u) = ∑**_{n=1}^{ꝏ}** (f**^{n}**(a)/n!) (u-a)**^{n}

f(u) = f(a) + f'(a)(u-a)/1! + f”(a)(u-a)^{2}/2! + f”'(a)(u-a)^{3}/3! + …

Step 3: Find the derivatives of the given function.

f’(u) = e^{u} f’’(u) = e^{u}

f’’’(u) = e^{u} f’’’’(u) = e^{u}

Step 4: put the given point in the above functions.

f’(0) = e^{0} = 1 f’’(0) = e^{0} = 1 f(0) = e^{0 }= 1

f’’’(0) = e^{0} = 1 f’’’’(0) = e^{0} = 1

Step 5: Put the above value carefully in the Taylor Series formula written in step 2 and simplify.

Where “a = 0”.

f(u) = f(a) + f'(a)(u-a)/1! + f”(a)(u-a)^{2}/2! + f”'(a) (u-a)^{3}/3! + …

f(u) = f(0) + f'(0)(u-0)/1! + f”(0)(u-0)^{2}/2! + f”'(0) (u-0)^{3}/3! + …

f(u) = 1 + 1(u-0)/1! + 1(u-0)^{2}/2! + 1 (u-0)^{3}/3! + …

f(u) = 1 + (u)/1! + (u)^{2}/2! + (u)^{3}/3! + …

**f(u) = 1 + (u)/1! + (u)**^{2}**/2! + (u)**^{3}**/3! + … is the Taylor series of “e**^{u }**” at “a = 0”.**

## Summary:

In this article, we discussed the definition of the Taylor series and its application. Furthermore, discussed the Taylor theorem in detail and solved the examples with full detail of trigonometric & exponential functions.