## Taylor Remainder Theorem

Recall that the *Taylor series* for a function about the point is given by

The *Taylor polynomial of degree * is the approximating polynomial which results from truncating the above infinite series after the degree term:

This is a good approximation for when is close to . How good an approximation is it? This module gives the answer to that question.

As in previous modules, let be the error between the Taylor polynomial and the true value of the function, i.e.,

Notice that the error is a function of . In general, the further away is from , the bigger the error will be.

## Taylor remainder theorem

The error is given precisely by

for some such that .

### Example

Consider the case when . The Taylor remainder theorem says that

for some such that . Solving for gives

for some , which is precisely the statement of the Mean value theorem. Therefore, one can think of the Taylor remainder theorem as a generalization of the Mean value theorem.

## Taylor error bound

As it is stated above, the Taylor remainder theorem is not particularly useful for actually finding the error, because there is no way to actually find the for which the equation holds. There is a slightly different form which gives a bound on the error:

where is the maximum value of over all in the interval .

### Example

Estimate using

and bound the error.

The function is , and the approximating polynomial used here is

Then according to the above bound,

where is the maximum of for . Since is an increasing function, . Thus,

Thus,