Truncation Error in Taylor Series

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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.

Contributed by: Vincent Shatlock and Autar Kaw (April 2011)
Open content licensed under CC BY-NC-SA


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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?

Reference

[1] A. K. Kaw, D. Nguyen, and E. E. Kalu, Numerical Methods with Applications, 2010. http://numericalmethods.eng.usf.edu/publications_book.html.



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