Study of Taylor Series: Definition, Theorem, Applications, & Examples

A Taylor series is a denoted function as an infinite sum of terms, where each term is a coefficient multiplied by the power of the independent variable. The series is named after the mathematician Brook Taylor and is a fundamental tool in calculus and mathematical analysis.

A mathematician Brook Taylor first presented the concept in the 18th century. The Taylor series is a dominant tool in mathematics and analysis that can be used to approximate functions, evaluate integrals, and explain differential equations.

The Taylor series can be used to estimate the value of a function at any point in its domain, and the correctness of the approximation depends on the number of positions involved in the series.

The concept of the Taylor series is an essential tool in modern mathematics and has led to numerous advances in analysis, physics, engineering, finance, number theory, and other fields. In this article, we will discuss the basic definition, theorem, application, advantages, and disadvantages.

What is the Taylor series?

The Taylor series is a mathematical concept that expresses a function of a polynomial with the infinite sum of terms involving its derivatives evaluated at a specific point. Each period of the Taylor polynomial series derives from the function's derivatives at a single point.

Formula :

The common formula for the Taylor series development of a function f(z) about a point z = a is given by :

f(z) = f(a) + f'(a)*(z-a) + f''(a)*(z-a) ^2/2! + f'''(a)*(z-a) ^3/3! + ...

Now, f'(a), f’’(a), f'''(a), ... represent the 1st, 2nd, 3rd, and higher derivatives of f(z) calculated at z=a, and the symbol n! denotes the factorial of n.

Theorem of Taylor series :

Taylor's series theorem is a mathematical theorem that offers a method for the rest period in the Taylor series of a function at a specific point. The English mathematician Brook Taylor supported the Taylor series conception in the 18th century, which is predictable for having this theorem remember his name.

The Taylor theorem conditions that several appropriately smooth functions f(z) can be approached by its nth-degree Taylor polynomial series at a specific point “a”, with the residue term Rn(z) assumed as:

f(z) = f(a) + f'(a)*(z-a)/1! + f''(a)*(z-a)2/2! + ... + fn(a)*(z-a) n/n! + Rn(z)

Where Rn(z) is the residue period, which is the same as the nth-derivative of f(z) calculated at particular point c between “a” and “z”, multiplied by the (z-c)(n+1) divided by (n+1)! which is

Rn(z) = [f(n+1) (c)*(z-c) (n+1)] / (n+1)!

The representation f(n)(z) symbolizes the nth derivative of f(z) concerning z.

Applications:

Taylor series is a mathematical tool used to approximate complex functions by expressing them as a sum of simpler functions, usually polynomials. It has many applications in different fields, including physics, engineering, finance, and computer science. Some of the common applications of the Taylor series are:

Approximation of Functions : Taylor series is used to approximate complex functions by expressing them as a sum of simpler functions, usually polynomials. This approximation is useful in many areas of engineering, such as control systems, signal processing, and data analysis.

Error Analysis : Taylor series can be used to estimate the error in a numerical method used to solve a problem. By comparing the Taylor series expansion of the exact solution with the numerical solution obtained using a method, we can estimate the accuracy of the method and improve its performance.

Optimization : Taylor series can be used to optimize functions by finding their critical points and inflection points. By expanding the function around these points, we can approximate it and determine its behavior in the neighborhood of the point.

Numerical Integration : Taylor series can be used to approximate the value of an integral by expressing the integrand as a Taylor series and integrating each term. This method is useful in many areas of physics and engineering where analytical solutions are not available.

Simulation : Taylor series can be used to simulate complex physical systems by approximating their behavior using simple models. This approach is used in many areas of physics, such as quantum mechanics and fluid dynamics, to simulate the behavior of complex systems.

Differential Equations : Taylor series can be used to solve differential equations by expressing the solution as a power series and finding the coefficients using the Taylor series expansion. This method is useful in many areas of physics and engineering where analytical solutions are not available.

Disadvantages and Advantages :

Here few advantages and disadvantages of Taylor series will be discussed.

Advantage :

The Taylor series is a mathematical tool that allows us to represent a wide range of functions as an infinite sum of terms that are related to the function's derivatives at a specific point. Here are some advantages of using the Taylor series:

Approximation : It is used to approximate the value of a function at a point p, especially for functions that are difficult or impossible to evaluate directly. By truncating the infinite series to a finite number of terms, we can get a close approximation to the true value of the function.

Derivative calculation : The Taylor series provides an efficient way to calculate the derivatives of a function at a specific point. Instead of using the limited definition of a derivative, which can be tedious and time-consuming, we can simply differentiate the terms of the Taylor series.

Analyzing function behavior : The Taylor series can be used to analyze the behavior of a function in the neighborhood of a specific point. By examining the terms of the series, we can determine the sign, relative size, and rate of change of the function's derivatives, which can provide valuable information about the function's behavior.

Note : Overall, the Taylor series is a powerful tool that has many advantages in the fields of mathematics, science, and engineering.

Disadvantage :

Here are some potential disadvantages of using the Taylor series to approximate functions :

Convergence issues : The Taylor series may not converge to the function at certain points, which can result in significant errors in the approximation.

Higher-order terms : Higher-order terms in the Taylor series can become increasingly complex, making the calculation of the approximation more difficult and time-consuming.

Difficulty in finding coefficients : In some cases, finding the coefficients of the Taylor series may be difficult, especially for functions with complex derivatives or those with no closed-form expression.

Note : Overall, the Taylor series can be a powerful tool for approximating functions, but they also have limitations that should be taken into account when using them.

Example section :

In this section, we explain the Taylor series with the help of examples.

Example 1 : The Taylor series expansion for the function f(z) = sin(z) centered at z=0:

Solution :

Step 1 :
f(z) = sin(z)
f(0) = sin(0) = 0  
 
Step 2 :
f'(z) = cos(z)
f'(0) = cos (0) = 1

Step 3 :
f''(z) = -sin(z)
f''(0) = -sin (0) = 0

Step 4 :
f'''(z) = -cos(z)
f'''(0) = -cos (0) = -1

Step 5 :
f''''(z) = sin(z)
f''''(0) = sin (0) = 0
...

Step 6 :
Using this information, we can write the Taylor series for sin(z) centered at z=0 as:
sin(z) = z - z^3/3! + z^5/5! - z^7/7! + ...

Step 7 :
The general formula for the Taylor series expansion of a function f(z) centered at z=a is :
f(z) = f(a) + f'(a)*(z-a)/1! + f''(a)*(z-a) ^2/2! + f'''(a)*(z-a) ^3/3! + ...
Where f', f'', f’’’, represent the 1st, 2nd, 3rd, etc. differentiation of f(z), correspondingly.

The above example of the Taylor polynomial can also be solved with the help of a Taylor series calculator to get rid of lengthy and time-consuming calculations.

Conclusion :

In this article, we have discussed the basic definition of the Taylor series, its theorem, application, advantages, and disadvantages of the Taylor series in detail. Additionally, discussed how to evaluate the Taylor series with the help of examples.  The Taylor series is a most interesting topic, hope you can determine the easily connected problem by understanding from this article.