Taylor and Maclaurin Series

 Learn All About the Taylor and Maclaurin Series

The Taylor and Maclaurin series are powerful mathematical tools for approximating functions using a series of terms. These series are named after the mathematicians, Brook Taylor and Colin Maclaurin, who made significant contributions to their development. Although the Taylor and Maclaurin series share many similarities, there are subtle differences between them, making each series useful in different contexts and practical uses. 

The Taylor Series 

The Taylor series is named after Brook Taylor, an English mathematician whointroduced the concept in the early 18th century. It is a mathematical expansion representing a function as an infinite sum of terms, where each term is obtained by differentiating the function at a specific point. Taylor’s idea was to approximate a function using a polynomial that could mimic its behavior near a specific point. 

The general form of a Taylor series is given by:

In this series, f(x) represents the original function, f(a) is the value of the function at the point a, f'(a) is the derivative of the function at a, f''(a) is the second derivative of the function at a, and so on. The terms involving the derivatives are multiplied by the corresponding powers of (x - a) divided by the factorial of the derivative order.

We can approximate a function with the Taylor series using a polynomial that closely matches the behavior of the original function near the point of expansion. We can also improve the accuracy of the approximation by including more terms in the series. The Taylor series is especially useful in mathematical analysis and calculus to be more specific, as it enables mathematicians to solve different equations by studying the properties of functions. 

The Maclaurin Series

The Maclaurin series is named after Colin Maclaurin, a Scottish mathematician who made significant contributions to calculus during the 18th century. It is a special case of the Taylor series and is obtained by expanding a function around the point a = 0, simplifying the general form of the Taylor series. 

The Maclaurin series has the form:

In this series, the terms involving the derivatives are evaluated at a = 0. The Maclaurin series allows for a simplified expression that closely approximates the function in the vicinity of the origin by expanding a function around this point. That makes it particularly useful when dealing with symmetric functions or when the point of interest is near zero.

The main difference between the Taylor and Maclaurin series is in the choice of the expansion point. The Taylor series allows for expansion around any point a, while the Maclaurin series specifically focuses on expansion around a = 0. Both series are invaluable tools in various fields of engineering and mathematics. They provide a means to approximate complex functions with simpler polynomial expressions, aiding in modeling, calculations, and analysis.

Conclusion

The Taylor and Maclaurin series are mathematical techniques allowing for the approximation of functions using polynomials. Understanding the development and subtle differences between these series allows engineers, mathematicians, and scientists to manipulate and analyze functions effectively, facilitating advancements in different fields of study.