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.


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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.
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?
[1] A. K. Kaw, D. Nguyen, and E. E. Kalu, Numerical Methods with Applications, 2010.
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