Calculus II · Unit 4B · lesson

Alternating-Series Approximation for Taylor Series

Concept

Learning objectives

use alternating-series error when it gives a simpler bound than Taylor's theorem.

Alternating-Series Approximation for Taylor Series

Explanation

Sometimes the coefficient pattern already contains the error estimate

Many Maclaurin series alternate with decreasing term magnitudes for the input being used. In that setting, the alternating-series remainder estimate bounds the error by the first omitted term. No global derivative bound is required, and the computation is often sharper and simpler.

The magnitudes must actually decrease at the chosen input. A series that is alternating symbolically may have early terms that grow before factorials take over. One can begin the estimate after monotonic decrease is established. As always, the first omitted term depends on the truncation, not merely on the polynomial degree label.

Bridge

Some Taylor errors are controlled by the next omitted term

When a Taylor series alternates with decreasing term magnitudes, the Alternating Series Estimation Theorem gives a sharper and simpler bound than the general Taylor remainder. The error is no larger than the first omitted term.

The conditions must be checked at the chosen input. A series may alternate symbolically while term magnitudes fail to decrease for a large input. Once the conditions hold, consecutive Taylor partial sums bracket the true value and even reveal whether the current approximation is above or below it.

Proof idea

This is the same shrinking-bracket theorem from Unit 4A

At a fixed input, the Taylor expansion becomes a numerical alternating series. Once magnitudes decrease to zero, its partial sums bracket the function value and the first omitted term bounds the error.

Concept

Alternating approximation error

If an alternating series has decreasing term magnitudes, then truncation error is at most the magnitude of the first omitted term.

Guided walkthrough

Approximate cosine

Using

\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots,

approximating $\cos(0.3)$ through the $x^4$ term has error at most

\frac{0.3^6}{6!}<1.02\times10^{-6}.

Worked example

Approximate cosine with a next-term guarantee

Use

\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots

to approximate $\cos(0.5)$ with three terms:

1-\frac{0.5^2}{2}+\frac{0.5^4}{24}=0.877604\ldots.

The first omitted term has magnitude

\frac{0.5^6}{6!}<0.000022.

Therefore the approximation error is below $2.2\times10^{-5}$ .

Common mistake

The next-term bound needs decreasing magnitudes

Do not quote the alternating bound from signs alone. Verify that the term magnitudes decrease toward zero at the selected input.

Interactive checku4b-alternating_series_approximation-01

What bounds the error after the $x^4$ term in $\cos(0.3)$ ?

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Show hint

Use the first omitted $x^6/6!$ term.

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Exercise

Approximate $\sin(0.4)$ with three nonzero terms and bound the error.

Exercise

Compare alternating and Lagrange bounds for cosine.

Exercise

Explain when early term magnitudes might not decrease.

Exercise

Choose a truncation guaranteeing six decimal places.

After the explanation

Use the section idea

Reading lens

Pair every Taylor approximation with a degree, center, target input, and certified remainder bound.

Mental model

The polynomial supplies the estimate; the remainder theorem supplies the trust boundary.

Decision

Choose a tractable center and degree, bound the needed derivative on the whole interval, then compare the bound with the required tolerance.

Common trap

Evaluating the next term without checking that the theorem's hypotheses make it a valid error bound.

Check yourself

Can you state exactly why the reported digits are certified?

Source & rights

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