What is M in the Lagrange error bound?

M is any upper bound for the absolute value of the (n+1)th derivative on the interval between the center and the point you're estimating. It doesn't have to be the exact maximum.

Is the Lagrange error bound the exact error?

No. It's a guarantee that the true error is no larger than the bound. The actual error is usually smaller.

When can I use the alternating series error bound?

When the series alternates in sign at the value of x you're using, and the size of the terms decreases to zero.

Lagrange and alternating series error bounds, explained

Updated November 9, 2026

When you estimate a value with a Taylor polynomial, an error bound tells you the most your estimate could be off. AP Calculus BC tests two of them: the Lagrange error bound and the alternating series error bound.

They answer the same question in different situations. This guide shows how each one works, how to choose the value $M$ , and how to tell which one you need.

Which bound should you use?

The series at that value of $x$ alternates, and its terms decrease to $0$ : use the alternating series bound. It's the quickest.
The question gives a bound on a derivative, or the terms don't alternate: use the Lagrange bound.

The alternating series error bound

If a series alternates and its terms decrease in size to $0$ , the error after stopping is at most the size of the first term you left out.

First omitted term

Use the first three nonzero terms of the Maclaurin series for

\ln(1 + x)

to approximate

\ln(1.2)

, and bound the error.

With $x = 0.2$ : $0.2 - \frac{0.2^2}{2} + \frac{0.2^3}{3} \approx 0.18267$ .

The terms alternate and decrease to $0$ , so the error is at most the next term: $\frac{0.2^4}{4} = 0.0004$ . The true value is about $0.18232$ , so the actual error is about $0.00035$ , inside the bound.

Answer: Error $\le 0.0004$

The Lagrange error bound

For a Taylor polynomial $P_n$ of degree $n$ centered at $a$ , the error at $x$ satisfies $|f(x) - P_n(x)| \le \frac{M}{(n+1)!}|x - a|^{n+1}$ . Here $M$ is any number at least as large as $|f^{(n+1)}|$ on the interval between $a$ and $x$ .

When a bound is given

Let

P_3

be the third-degree Taylor polynomial for

f

about

x = 1

, and suppose

|f^{(4)}(x)| \le 6

for

1 \le x \le 2

. Find an upper bound for

|f(1.5) - P_3(1.5)|

With $M = 6$ , $n = 3$ , and $|x - a| = 0.5$ : $\frac{6}{4!}(0.5)^4 = 0.015625$ .

Answer: $0.015625$

Choosing M yourself

Use the third-degree Maclaurin polynomial for

e^x

to approximate

e^{0.5}

, and find a Lagrange error bound.

$P_3(0.5) = 1 + 0.5 + \frac{0.5^2}{2} + \frac{0.5^3}{6} \approx 1.64583$ .

Every derivative of $e^x$ is $e^x$ , which on $[0, 0.5]$ is largest at $x = 0.5$ . Since $e^{0.5} < 2$ , take $M = 2$ . A simple overestimate is fine, as $M$ only needs to be large enough.

The bound is $\frac{2}{4!}(0.5)^4 \approx 0.00521$ . The actual error is about $0.00289$ , which is less, as it must be.

Answer: Error $\le 0.00521$

When both bounds work

Sometimes either bound applies, and you can use whichever is simpler.

Sine at a small angle

Use

x - \frac{x^3}{6}

to approximate

\sin(0.5)

, and bound the error.

The estimate is $0.5 - \frac{0.125}{6} \approx 0.47917$ . The Maclaurin series for sine alternates at $x = 0.5$ , and its terms decrease, so the error is at most the next term: $\frac{0.5^5}{5!} \approx 0.00026$ .

The true value is about $0.47943$ , so the actual error of about $0.00026$ is just inside the bound.

Answer: Error $\le 0.00026$

Common mistakes

Using derivative $n$ instead of $n + 1$ . The bound always uses one more than the polynomial's degree.
Finding $M$ at the center only. $M$ must work on the whole interval between $a$ and $x$ .
Skipping the conditions. For the alternating bound, say that the terms alternate and decrease to $0$ .
Counting zero terms. The polynomial for sine of degree 3 has no $x^2$ term. Go by degree when using Lagrange, and by terms actually left out when using the alternating bound.

To review the polynomials these bounds apply to, see the Taylor series guide.

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Questions

What is M in the Lagrange error bound?: M is any upper bound for the absolute value of the (n+1)th derivative on the interval between the center and the point you're estimating. It doesn't have to be the exact maximum.
Is the Lagrange error bound the exact error?: No. It's a guarantee that the true error is no larger than the bound. The actual error is usually smaller.
When can I use the alternating series error bound?: When the series alternates in sign at the value of x you're using, and the size of the terms decreases to zero.