Examples

Taylor polynomial approximations

Let and let be the Taylor polynomials expanded around .

By considering the alternating series error bound, find the first for which must have error less than .

Solution

Write the Maclaurin series of because we are expanding around :

This series is alternating, so the AST error bound formula applies (“Next Term Bound”):

Find smallest such that , and then we know:

Plug in the series for :

Solve for the first time by listing the values:

The first time is below happens when .

This is NOT the same as in . That is the highest power of allowed.

The sum of prior terms is .

Since because there is no term, the final answer is .

Taylor polynomials to approximate a definite integral

Approximate using a Taylor polynomial with an error no greater than .

Solution

Plug into the series of :

Find an antiderivative by terms:

Plug in bounds for definite integral:

Notice alternating series pattern. Apply error bound formula, “Next Term Bound”:

So we can guarantee an error less than by summing the first terms through :