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The Remainder Term in Taylor Series

We can express an infinitely differentiable functionin terms of the associated Taylor series:

whereand(if with the obvious similar result for)

is the remainder term and allows us to estimate the accuracy of the Taylor series on an interval.

Proof: Taylors theorem is derived using integration by parts repeatedly:

Where

We can use the mean value theorem to writewhere ifwith the obvious similar result for

The Taylor term can be used in the following way:

Estimate the size of the error foron the interval

The differentials ofgoetc and sinceis increasing on the intervalon this interval hence