Truncation Error in Taylor Series
Taylor series for transcendental functions have an infinite number of terms. This Demonstration shows the truncation error created by using a finite number of terms in approximating three such functions with Taylor series based at zero.
The general form of a Taylor series is
assuming the function and all its derivatives exist and are continuous on an interval centered at and containing .
Here are the Maclaurin series (a special case of a Taylor series written around the point ) for the three functions considered:
In this Demonstration, we show the truncation error as a function of the number of terms of the Maclaurin series for the particular value of the function argument.
Questions: 1. For what values of would the given Maclaurin series for diverge? 2. How would you choose the number of terms to get the value of correct up to a specified number of significant digits? 3. The other source of error in numerical methods is round-off error. What influence does this have on the accuracy of the approximations?
 A. K. Kaw, D. Nguyen, and E. E. Kalu, Numerical Methods with Applications, 2010. http://numericalmethods.eng.usf.edu/publications_book.html.