

When using a Taylor polynomial of degree n centered at c to approximate the value of a function f at x, there is an error because the polynomial does not exactly mimic the function (unless, of course, f is a polynomial of degree less than or equal to n). We can bound this error using the Lagrange remainder (or Lagrange error bound). The remainder is:
where M is the maximum of the absolute value of the (n + 1)th derivative of f on the interval from x to c. The error is bounded by this remainder (i.e., the absolute value of the error is less than or equal to R). Note that R depends on how far x is away from c, how big n is, and on the characteristics of f.
See About the calculus applets for operating instructions. 

The applet shows the Taylor polynomial with n = 3, c = 0 and x = 1 for f (x) = e^{x}. To compute the Lagrange remainder we need to know the maximum of the absolute value of the 4th derivative of f on the interval from 0 to 1. Since the 4th derivative of e^{x} is just e^{x}, and this is a monotonically increasing function, the maximum value occurs at x = 1 and is just e. So:
Note in the applet that the actual error is about 0.052. The Lagrange remainder is a bound on the error, not the actual error itself. It just says that the error, whatever it is, will be less than the Lagrange remainder.
Other 'Sequences and Series' topics
Acknowledgements
Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.

