Taylor series is a very important concept that is used in numerical methods. From the concept of truncation error to finding the true error in Trapezoidal rule, having a clear understanding of Taylor series is extremely important. Other places in numerical methods where Taylor series concept is used include: the derivation of finite difference formulas for derivatives, finite difference method of solving differential equations, error in Newton Raphson method of solving nonlinear equations, Newton divided difference polynomial for interpolation, etc.

I have written a short chapter on Taylor series. After reading the chapter, you should be able to:

1. understand the basics of Taylor’s theorem,

2. see how transcendental and trigonometric functions can be written as Taylor’s polynomial,

3. use Taylor’s theorem to find the values of a function at any point, given the values of the function and all its derivatives at a particular point,

4. errors and error bounds of approximating a function by Taylor series,

5. revisit the chapter whenever Taylor’s theorem is used to derive or explain numerical methods for various mathematical procedures.

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I feel that some of your example (Example 5 of this section) make mathematical leaps of logic that not all students follow. Your calculations for true error and the remainder do not have clear steps to express where each variable is originating, nor do your examples correspond to either of the calculus textbooks I have as they calculate error differently for these series. It would be lovely if you included more details as explanation so that the text was a better learning tool.

I feel that some of your example (Example 5 of this section) make mathematical leaps of logic that not all students follow. Your calculations for true error and the remainder do not have clear steps to express where each variable is originating, nor do your examples correspond to either of the calculus textbooks I have as they calculate error differently for these series. It would be lovely if you included more details as explanation so that the text was a better learning tool.

This is only a primer as a refresher for prerequisite courses. Here is a reference. http://www.millersville.edu/~bikenaga/calculus/remainder-term/remainder-term.pdf