Choosing And Verifying An Error-Bound Strategy

Choosing a Bound Is Part of the Approximation

A Taylor polynomial gives a computable approximation, but the polynomial alone does not show how accurate that approximation is. To certify an answer, you need an error bound that applies to the particular function, center, interval, and polynomial in the problem.

Two useful choices are the Lagrange remainder bound, which uses a bound on a derivative, and the alternating-series remainder bound, which uses the sizes and signs of the terms in a convergent alternating series. Neither is automatically available just because a Taylor polynomial has been written down. The conditions must be checked before the bound is used.

The Two Error-Bound Strategies

The Lagrange form of the Taylor remainder says that if \(f\) has continuous derivatives through order \(n+1\) on the interval between \(a\) and \(x\), then for some \(\xi\) between \(a\) and \(x\), $$ f(x)-P_n(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}. $$ Consequently, if \(\lvert f^{(n+1)}(t)\rvert\le M\) throughout that interval, then $$ \lvert f(x)-P_n(x)\rvert\le \frac{M\lvert x-a\rvert^{n+1}}{(n+1)!}. $$ The derivative bound must hold everywhere on the interval, not just at the expansion center or at the target.

The alternating-series bound applies when the function is represented at the target by a convergent alternating series whose term magnitudes decrease to zero. If the first omitted term has magnitude \(b\), the absolute error after the retained terms is at most \(b\). To use this rule, verify that the series converges at the target, that its signs alternate, and that its term magnitudes decrease from the omitted term onward.

These methods answer the same practical question but use different information. A derivative bound can work without an alternating series, while an alternating-series estimate can be much sharper when its conditions hold. A valid strategy is the one whose hypotheses can be verified and whose bound is small enough for the requested tolerance.

Why the Alternating-Series Bound Works

Here is a short justification of the error estimate. Suppose the omitted tail has alternating terms with magnitudes \(b_m,b_{m+1},\ldots\), where \(b_m\ge b_{m+1}\ge\cdots\ge0\) and \(b_k\to0\). First suppose the first omitted term is positive. Grouping consecutive terms gives $$ b_m-b_{m+1}+b_{m+2}-b_{m+3}+\cdots. $$ Each pair \(b_{m+2j}-b_{m+2j+1}\) is nonnegative, so the tail is nonnegative. Also, after its first term, the remainder can be grouped as $$ -b_{m+1}+b_{m+2}-b_{m+3}+\cdots, $$ which is nonpositive by the same pairing argument. Thus the full tail lies between \(0\) and \(b_m\). The argument for a negative first omitted term follows by changing signs. Taking limits of the finite partial sums is justified by the assumed convergence. Therefore the tail has the sign of the first omitted term and magnitude at most that term.

A Practical Decision Process

For a mixed set of approximation tasks, use a consistent sequence of checks rather than choosing a method by habit.

Worked Example: Choosing an Alternating Bound for Sine

Worked Example: Using a Lagrange Bound for the Exponential

Worked Example: Checking an Alternating Logarithm Series

A Common Degree-Selection Mistake

A frequent error is to skip a nonzero term when identifying the first omitted term. For example, the degree-5 Taylor polynomial for \(\sin(0.5)\) is $$ P_5(0.5)=0.5-\frac{(0.5)^3}{3!}+\frac{(0.5)^5}{5!}. $$ One might incorrectly claim that its alternating-series error is at most \((0.5)^9/9!\), by looking at the degree-7 polynomial used in the earlier example. But the degree-5 polynomial omits the degree-7 term first. Its valid alternating-series bound is $$ \left|\sin(0.5)-P_5(0.5)\right| \le\frac{(0.5)^7}{7!} =\frac{0.0078125}{5040} =0.0000015500992\ldots, $$ which is greater than \(10^{-6}\). The claimed degree-5 certificate therefore fails. To meet this tolerance using the alternating bound, retain the degree-7 term and bound the error by the degree-9 term. Always identify the first omitted nonzero term of the polynomial actually used.

How to Present a Complete Certificate

A clear certificate should make the reasoning auditable. State the target and tolerance, define the polynomial being evaluated, and name the bound used. Then verify the hypotheses: continuity and a derivative maximum or bound for Lagrange; convergence, alternating signs, and decreasing magnitudes for the alternating-series rule. Finally, display the numerical inequality that compares the valid error bound with the tolerance.

Keep the exact polynomial distinct from its decimal display. A rounded polynomial value is useful for reporting an approximation, but the error theorem bounds the difference between the function value and the exact polynomial value. If a rounded output itself needs to be certified, the rounding error must also be included in the total error. In the sine example, for instance, the rigorous statement compares \(\sin(0.5)\) with \(P_7(0.5)\), not with a shortened decimal that might shift the difference.

The two methods may both apply, and one may provide a tighter bound. It is legitimate to compare them, but only after checking both sets of conditions. A failed or oversized estimate is not proof that the approximation is inaccurate; it means only that this bound does not certify the desired tolerance. Try a higher degree, sharpen a justified derivative bound, or use another method whose conditions hold.