Why Compare Error Bounds?
A Taylor polynomial gives a manageable approximation to a function, but an approximation is more useful when you can also say how far it may be from the true value. Two common tools are the Lagrange remainder estimate and the alternating-series remainder estimate. They can apply to the same approximation, but they use different information and may give different bounds.
The important first step is not to calculate. It is to check the conditions for each method. A small-looking bound is not justified if the method’s hypotheses do not hold.
The Two Error Estimates
Let \(P_n(x)\) be the Taylor polynomial of degree \(n\) for \(f\), centered at \(a\):
The error, or remainder, at \(x\) is \(R_n(x)=f(x)-P_n(x)\). The two estimates bound the absolute error \(\lvert R_n(x)\rvert\).
For the alternating-series estimate, the Taylor series must actually represent \(f(x)\). Its terms after the polynomial must alternate in sign, have nonincreasing magnitudes, and tend to zero. Under those conditions, the alternating-series remainder theorem bounds the error by the magnitude of the first omitted nonzero term.
These hypotheses are not interchangeable. The Lagrange estimate requires a bound on a derivative over the entire interval. The alternating estimate requires an appropriate alternating series and decreasing term magnitudes at the point being evaluated.
A Reliable Comparison Process
Identify the center, evaluation point, and Taylor polynomial degree. Both estimates must refer to this same approximation.
Verify the needed derivative exists and is continuous on the interval. Find a valid upper bound \(M\) for its absolute value there.
Confirm the Taylor series represents the function at the evaluation point, and check alternating signs, decreasing magnitudes, and convergence of the terms to zero.
Calculate each applicable estimate. A bound is useful only if its conditions have been established.
Worked Example: Both Methods Apply to a Cosine Approximation
Worked Example: Comparing Two Bounds for \(\cos(1)\)
Approximate \(\cos(1)\) with the degree-two Maclaurin polynomial, then compare a Lagrange bound with an alternating-series bound.
The polynomial is
At \(x=1\),
Lagrange estimate. The third derivative is \(f'''(x)=\sin x\). On \([0,1]\), \(\lvert\sin x\rvert\le 1\), so we can take \(M=1\). The estimate gives
The derivative is continuous on the interval, so Taylor’s theorem applies.
Alternating-series estimate. The cosine series at \(x=1\) begins
The terms alternate, and their magnitudes decrease to zero: the ratio of consecutive nonzero magnitudes is \(1/((2k+2)(2k+1))\), which is less than \(1\) for \(k\ge 0\). The first omitted nonzero term after \(P_2\) is \(1/4!=1/24\). Thus
Since \(1/24<1/6\), the alternating-series estimate is sharper here. Both estimates are valid; the difference is that the Lagrange bound uses a broad derivative bound, while the alternating estimate uses the size of the next series term.
Check: The approximation is \(1/2\), and the stated error bounds are \(1/6\) and \(1/24\), respectively. The sharper estimate still bounds the error relative to that same polynomial.
Why the Derivative Bound Must Cover the Whole Interval
The Lagrange formula involves the derivative at an unknown point \(c\) between the center and the evaluation point. You do not get to choose \(c\), so a value of the derivative at just the center or at the endpoint is not enough. Instead, find a number \(M\) that bounds the absolute derivative everywhere on the interval.
The alternating-series method uses different information: the size of the first omitted term. When the next nonzero term occurs at degree \(n+1\), that term is \(\lvert f^{(n+1)}(a)\rvert\lvert x-a\rvert^{n+1}/(n+1)!\). The Lagrange bound is at least this large, since its \(M\) must bound the derivative at \(a\) as well. The alternating-series theorem then bounds the actual error by that first omitted term. If some Taylor coefficients vanish, the first omitted nonzero term may come later; calculate the two bounds rather than relying on this comparison.
Worked Example: The Estimates Can Agree
Worked Example: Comparing Bounds for \(\ln(1.5)\)
Use the degree-one Maclaurin polynomial for \(f(x)=\ln(1+x)\) at \(x=1/2\). Compare both error bounds.
Since \(f(0)=0\) and \(f'(0)=1\), the polynomial is \(P_1(x)=x\). Therefore,
Lagrange estimate. The second derivative is \(f''(x)=-1/(1+x)^2\). On \([0,1/2]\), its absolute value is at most \(1\), so \(M=1\) is valid. Thus
The function has a continuous second derivative throughout this interval.
Alternating-series estimate. The Taylor series is
At \(x=1/2\), the term magnitudes \( (1/2)^k/k\) decrease to zero, and the signs alternate. The first omitted term after \(P_1\) has magnitude
So the alternating-series estimate is also \(1/8\). Here the two valid estimates agree.
Check: The polynomial value is \(1/2\), and both methods give the same bound, \(1/8\), for its error.
Worked Example: When the Alternating Estimate Does Not Apply
Worked Example: Using Lagrange for \(\ln(0.5)\)
Use the degree-two Maclaurin polynomial for \(\ln(1+x)\) at \(x=-1/2\). Check whether each method applies.
The polynomial is \(P_2(x)=x-x^2/2\), so
Lagrange estimate. The third derivative is \(f'''(x)=2/(1+x)^3\). On \([-1/2,0]\), it is positive and increases as \(x\) decreases, so its largest absolute value is \(f'''(-1/2)=2/(1/2)^3=16\). We can take \(M=16\). The function has a continuous third derivative on the interval. Therefore,
Alternating-series check. Substitution into the logarithmic series gives terms \(-1/2,-1/8,-1/24,\ldots\). They all have the same sign, so the alternating-series remainder theorem does not apply. The Lagrange bound is valid; no alternating-series bound has been justified.
Check: The actual error is \(\lvert\ln(1/2)+5/8\rvert=\ln 2-5/8\approx0.0681\), which is less than \(1/3\), as the Lagrange estimate guarantees.
A Common Mistake: Using the Next Term Without Checking the Signs
For the last example, a tempting but incorrect calculation is to call \(1/24\) an alternating-series error bound just because it is the next term’s magnitude. That would claim an error at most \(1/24\approx0.0417\). But the series terms at \(x=-1/2\) do not alternate, so the alternating-series theorem gives no such conclusion. In fact, the error is approximately \(0.0681\), which is greater than \(1/24\). The correction is to check the signs before using the theorem, then use the valid Lagrange estimate \(1/3\) in this example.
Another common mistake is to find the largest derivative value only at the evaluation point. The Lagrange estimate requires a bound across the entire interval between the center and that point. If the derivative varies, an endpoint or interior value may be larger than the value at the point of evaluation.
How to Choose in Practice
- Start with the alternating-series check when the Taylor expansion has alternating terms. If its hypotheses hold, the first omitted term often gives a sharp bound.
- Use Lagrange when it applies, including when the Taylor terms do not alternate. It requires a derivative bound on the interval, not a particular pattern in the series coefficients.
- When both apply, calculate both. Use the smaller valid upper bound. Do not assume that one method is always sharper, especially when zero Taylor coefficients cause a gap before the first omitted nonzero term.
- Keep the polynomial fixed. Bounds for different truncation degrees do not compare the error of the same approximation.
These estimates certify accuracy: they show that the true value lies within a specified distance of the polynomial value. A rigorous comparison therefore rests on two things—correct arithmetic and verified hypotheses.
Check Your Understanding
For each question, identify which estimates are justified before comparing their bounds.
- What derivative information is needed for the Lagrange remainder estimate?
- What three term conditions are needed for the alternating-series remainder estimate?
- For \(\cos(1)\) approximated by \(P_2(1)\), which of the two bounds in the worked example is smaller?
- Why can the first omitted term not automatically be used as an error bound for \(\ln(1-1/2)\)?
- If both estimates apply to the same polynomial, what should you do before choosing one?
