Calculus II · Unit 4B · lesson
Alternating-Series Approximation for Taylor Series
Apply alternating-series error estimates to Taylor approximations and determine a sufficient polynomial degree.
Section overview
Taylor bounds and approximationWhat this section is building
Apply alternating-series error estimates to Taylor approximations and determine a sufficient polynomial degree.
The polynomial supplies the estimate; the remainder theorem supplies the trust boundary.
Choose a tractable center and degree, bound the needed derivative on the whole interval, then compare the bound with the required tolerance.
Evaluating the next term without checking that the theorem's hypotheses make it a valid error bound.
Learning objectives
use alternating-series error when it gives a simpler bound than Taylor's theorem.
Alternating-Series Approximation for Taylor Series
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.
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.
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.
Alternating approximation error
If an alternating series has decreasing term magnitudes, then truncation error is at most the magnitude of the first omitted term.
Approximate cosine
Using
approximating through the term has error at most
Approximate cosine with a next-term guarantee
Use
to approximate with three terms:
The first omitted term has magnitude
Therefore the approximation error is below .
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.
u4b-alternating_series_approximation-01What bounds the error after the term in ?
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Use the first omitted term.
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Approximate with three nonzero terms and bound the error.
Compare alternating and Lagrange bounds for cosine.
Explain when early term magnitudes might not decrease.
Choose a truncation guaranteeing six decimal places.
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