Remainders and Error Bounds

An approximation needs an accuracy claim

A Taylor polynomial is exact at its center and matches several derivatives there, but a finite polynomial leaves out infinitely many later terms. The difference between the function and its degree-$N$ Taylor polynomial is the remainder:

$$R_N(x)=f(x)-T_N(x).$$

Writing an approximation without saying anything about this remainder is incomplete when accuracy matters. A decimal result may look plausible, yet a model should state how close it is expected to be and why. Error bounds turn local derivative information into a guarantee rather than a hopeful estimate.

Taylor’s inequality says that if $|f^{(N+1)}(z)|\le M$ for every $z$ between the center $a$ and target $x$, then

$$|R_N(x)|\le\frac{M}{(N+1)!}|x-a|^{N+1}.$$

Every factor has a role. The derivative bound $M$ measures how large the first omitted type of behavior can be. The power of $|x-a|$ records distance from the center. The factorial in the denominator can make the bound shrink quickly as degree increases. The statement is a bound, not generally the exact error.

Bound a familiar exponential approximation

Approximate $e^{0.2}$ with the quadratic Maclaurin polynomial

$$T_2(x)=1+x+\frac{x^2}{2}.$$

For $f(x)=e^x$, every derivative is $e^x$. On the interval from $0$ to $0.2$, the third derivative is at most $e^{0.2}$. A simple bound such as $M=2$ is valid because $e^{0.2}<2$. Taylor’s inequality gives

$$|R_2(0.2)| \le\frac{2}{3!}(0.2)^3 =\frac{0.016}{6} \approx0.00267.$$

The polynomial evaluates to $1.22$, so the true value lies within about $0.00267$ of $1.22$. The bound is deliberately a little loose because $M=2$ was easy to verify. A tighter bound could use $M=e^{0.2}$, but a valid simple guarantee is often more useful than an exact-looking expression that has not been justified.

The interval between $a$ and $x$ is essential. Choosing $M$ only from the derivative value at the center can fail if the derivative grows on the path to the target. For $e^x$ and a positive target, the maximum occurs at the target; for a negative target, it occurs at zero. Sketching the interval prevents an unexamined maximum claim.

It is also important to preserve the absolute value in the error statement. A Taylor polynomial may lie above or below the true function depending on the function, degree, and target input. The inequality bounds the magnitude of the difference and does not by itself name its sign. If the sign matters, it must be established from additional information such as the first omitted term in an alternating series or a known derivative sign. Accuracy and direction are related but separate conclusions.

Alternating-series error is a different tool

When a Taylor series has alternating terms whose magnitudes decrease to zero at the target, the alternating-series estimate can be simpler. The error after a truncation is at most the magnitude of the first omitted term. For the sine series at $x=0.3$,

$$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots,$$

using $x-x^3/3!$ gives an error no larger than $0.3^5/5!$. This test relies on the alternating, decreasing structure at the chosen input; it is not a replacement for Taylor’s inequality in every problem. State which theorem supports the bound rather than calling every omitted term “the error.”

The two estimates often have different strengths. Taylor’s inequality works from a derivative maximum and can apply when signs do not alternate. The alternating estimate can be sharp and easy when its hypotheses hold. Choosing a method is part of mathematical judgment: use the information the representation actually provides instead of forcing a favorite formula.

Do not confuse an upper bound with a decimal rounding rule. If an error bound is less than $0.0005$, then rounding a positive approximation to three decimal places can be justified under the usual nearest-rounding convention. If the bound is only less than $0.005$, it does not automatically justify three decimal places. Translate the requested tolerance into an inequality before selecting the degree. This makes the number of retained terms a consequence of the desired reliability, not an aesthetic choice.

Select a degree from a tolerance

An error tolerance can determine the polynomial degree. To approximate $e^{0.2}$ within $0.0001$, find an $N$ for which

$$\frac{2(0.2)^{N+1}}{(N+1)!}<0.0001.$$

Try successive degrees or compare the bound’s factors systematically. The point is not to find the smallest possible $N$ unless asked. Any degree that proves the required inequality is sufficient. Report the polynomial used and the bound obtained; a bare decimal with many digits does not explain why those digits should be trusted.

For retrieval, use Taylor’s inequality to bound the error in approximating $\cos(0.4)$ with its degree-two Maclaurin polynomial. Choose a valid upper bound for the third derivative on the relevant interval, even if it is not tight. Then compare with the alternating-series estimate and explain which is sharper here. The next lesson combines series operations, approximation, and error judgment in modeling choices rather than treating them as isolated techniques.