The Remainder Term in Taylor Series
We can express an infinitely differentiable function
in terms of the associated Taylor series:
where
and
(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 write
where 
ifwith the obvious similar result for
The Taylor term can be used in the following way:
Estimate the size of the error for
on the interval
The differentials of
go
etc and sinceis increasing on the interval
on this interval hence