Differences Between the Taylor and Maclaurin Series

Taylor vs Maclaurin Series

Aside from flying cockroaches, here is another thing that most people detest – math. We are often stricken with fear when we are facing math. The numbers seem like they are rattling our head, and it seems that math is eating up all of our life force. No matter what we do, we can’t escape the clutches of math. From counting to complex equations, we are always dealing with math. Nevertheless, we have to deal with it. Face your fear and learn to handle it. We have to meet Taylor and Maclaurin. Who are these people? These are not people. These are mathematical series.

In the field of mathematics, a Taylor series is defined as the representation of a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point. The Taylor series got its name from Brook Taylor. Brook Taylor was an English mathematician in 1715. It is all right to approximate the value of a function through utilizing the finite number of terms in the Taylor series. Approximating the value is already a common practice. In this approximation process, the Taylor series can yield quantitative estimates on the error. A Taylor polynomial is the term used to represent the finite number of the Taylor series’ initial function terms.

According to wikipedia.org, there are other uses of the Taylor series for determining analytic functions. The Taylor series can be used in obtaining the partial sums or the Taylor polynomials through using approximation techniques in the entire function. Another usage of the Taylor series is the differentiation and integration of the power series which can be done with each term. The Taylor series can also provide a complex analysis through integrating the analytic function with a holomorphic function in a complex plane. It can also be used to obtain and compute values numerically in a truncated series. This is done by applying the Chebyshev formula and Clenshaw algorithm. Another thing is that you can use the Taylor series in algebraic operations. An example of this is applying the Euler’s formula connecting with the Taylor series for the expansion of trigonometric and exponential functions. This can be used in the field of harmonic analysis. You can also use the Taylor series in the field of physics.

A Taylor series becomes a Maclaurin series if the Taylor series is centered at the point of zero. The Maclaurin series is named after Colin Maclaurin. Colin Maclaurin was a Scottish mathematician who had greatly used the Taylor series during the 18th century. A Maclaurin series is the expansion of the Taylor series of a function about zero. According to mathworld.wolfram.com, the Maclaurin series is a type of series expansion in which all terms are non-negative integer powers of the variable. Other more general types of series include the Laurent series and the Puiseux series. The Taylor and Maclaurin series have many uses in the mathematical field including the sciences.

Summary:

1. In the field of mathematics, a Taylor series is defined as the representation of a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point.

2. A Taylor series becomes a Maclaurin series if the Taylor series is centered at the point of zero. A Maclaurin series is the expansion of the Taylor series of a function about zero.

3. The Taylor series got its name from Brook Taylor. Brook Taylor was an English mathematician in 1715. The Maclaurin series is named after Colin Maclaurin. Colin Maclaurin was a Scottish mathematician who had greatly used the Taylor series during the 18th century.