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

(1) Write the Maclaurin series of because we are expanding around .

Alternating sign, odd function:

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(2)

Notice this series is alternating, so AST error bound formula applies.

AST error bound formula is:

Here the series is and is the error.

Notice that is part of the terms in this formula.

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(3) Implement error bound to set up equation for .

Find such that , and therefore by the AST error bound formula:

Plug in .

From the series of we obtain for :

We seek the first time it happens that .

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(4) Solve for the first time .

Equations to solve:

Method: list the values:

The first time is below happens when .

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(5) Interpret result and state the answer.

When , the term at is less than .

Therefore the sum of prior terms is accurate to an error of less than .

The sum of prior terms equals .

Since because there is no term, the same sum is .

The final answer is .

It would be wrong to infer at the beginning that the answer is , or to solve to get .

Taylor polynomials to approximate a definite integral

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

Solution

(1) Write the series of the integrand.

Plug into the series of :

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(2) Compute definite integral by terms.

Antiderivative by terms:

Plug in bounds for definite integral:

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(3) Notice AST, apply error formula.

Compute some terms:

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

Final answer is .