ISPA - Taylor and Maclaurin Series

Taylor and Maclaurin Series

Maclaurin and Taylor Series

image of Brook Taylor image of Colin Maclaurin Image Caption (to the left): Brook Taylor (1685-1731) was an English mathematician who added to mathematics a new branch called the 'calculus of finite differences', invented integration by parts, and discovered the celebrated formula known as Taylor's expansion. Taylor was elected a fellow of the Royal Society of London in 1712 and in the same year sat on the committee for adjudicating Sir Isaac Newton's and Gottfried Wilhelm Leibniz's conflicting claims of priority in the invention of calculus.'

Image Caption (to the right): Colin Maclaurin (1698-1746), a 17^th century Scottish mathematician who is best known for his work on approximating functions by series expanded around 0, was not the first do so. Taylor, Newton, Gregory, Bernoulli, and Madhava in 14^th century India, were all known to have done work with series expansion. When Maclaurin published his work in Methodus incrementorum directa et inversa, he did give credit to Brook Taylor for his contribution.

A Maclaurin series is a power series for any function f with infinitely many derivatives throughout some interval containing 0 as an interior point and of the form equation image indicator $LaTeX: \sum_{n=0}^\infty \frac{f^{\left(n\right)}\left(0\right)}{n!}\left(x^n\right)$ $\sum_{n=0}^\infty \frac{f^{\left(n\right)}\left(0\right)}{n!}\left(x^n\right)$ . Although this definition is conveniently centered about x = 0, there are other situations where a different center is needed. A power series for an infinitely differentiable function throughout some interval containing c as an interior point and of the form equation image indicator $LaTeX: \sum_{n=0}^\infty \frac{f^{\left(n\right)}\left(c\right)}{n!}\left(x-c\right)^n$ $\sum_{n=0}^\infty \frac{f^{\left(n\right)}\left(c\right)}{n!}\left(x-c\right)^n$ , where f⁽ⁿ⁾(c) is the nth derivative of f at c is a Taylor Series. Note that the Maclaurin Series is a special case of the Taylor series.

Maclaurin Series for cos x, sin x, and e^x

Many familiar functions can be expressed as Maclaurin series. The next two presentations illustrate how to express the cosine and sine functions as Maclaurin series.

It may be helpful to see a visualization of how well the Maclaurin series approximates the sine function.

Another familiar function f(x) e^x is often expressed as a Maclaurin series, as the video below explains.

As before, a visualization of how well a Maclaurin series approximates the e^x function follows:

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Taylor Series

Although expressing functions as Maclaurin series is often convenient, using a Taylor series is necessary when functions are not defined at x = 0, such as f(x) = ln x or equation image indicator $LaTeX: f\left(x\right)=\frac{1}{\sqrt[]{x}}$ $f\left(x\right)=\frac{1}{\sqrt[]{x}}$ . View the presentation below beginning at 8:02 that uses a Taylor series to represent a function not centered at x = 0.

An investigation of the generalized Taylor series approximation is the focus of the next video.

You may find the next presentation focusing on writing the Taylor series of a specific polynomial to be very enlightening.

The next three presentations involve approximating the non-algebraic functions e^x, cos x, and sin x using Maclaurin Series.

Manipulation of Taylor Series to Form New Series

The substitution and multiplication techniques used in a previous lesson for generating a new series from a known geometric series can be applied to Taylor series as well. The presentation below shows how substitution and finding the product of two series can be used to generate new Taylor series.

The next video illustrates how finding the quotient of two power series generates a new Taylor series.

Differentiation and Integration

Likewise, the techniques of differentiation and integration may be effectively employed to generate new Taylor series. Earlier in this module, the functions e^x, sin x, and cos x were expressed as power series. The next presentation uses differentiation and integration of these power series to generate other Taylor series.

Power series are also useful in approximating definite integrals for which no elementary formula exists. Consider the following example:

Power series examplel in approximating definite integrals

Taylor and Maclaurin Series Practice

Taylor and Maclaurin Series: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of Taylor and Maclaurin series.

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