Alternating Series Error Bound

AP Calculus BC· AP Calculus BC CED — Infinite Sequences and Series· 14 min read

1. Theorem Statement and Required Conditions★★☆☆☆⏱ 4 min

Alternating series error bound is a simple, powerful theorem that estimates the maximum possible error when you approximate the sum of a convergent alternating series with a finite partial sum. It is explicitly required by the AP Calculus BC CED, and appears regularly in both multiple-choice and free-response sections, often paired with Taylor polynomial approximation.

📘 Definition

Alternating Series Remainder Theorem

$∣ R_{n} ∣ = ∣ S - S_{n} ∣ \leq a_{n + 1}$

For an alternating series meeting two conditions from the Alternating Series Test, the magnitude of the error when approximating the exact sum $S$ with the $n$ th partial sum $S_{n}$ is at most the magnitude of the first neglected term $a_{n + 1}$ .

To apply the theorem, first write any alternating series in the standard form:

\sum_{k=1}^{\infty} (-1)^{k+1} a_k \quad \text{where } a_k > 0 \text{ for all } k

$lim_{k \to \infty} a_{k} = 0$ (the sequence of positive terms approaches zero)
The sequence ${a_{k}}$ is strictly decreasing for all $k \geq n$

Intuitively, partial sums of a convergent alternating series oscillate around the exact sum, so the error after stopping at $S_{n}$ can never exceed the size of the next (first neglected) term.

📐 Worked Example

Consider the alternating series $\sum_{k = 1}^{\infty} \frac{( - 1 ) ^{k + 1}}{k ^{2}}$ . Does the alternating series error bound apply to an approximation using the first 4 terms? If yes, what is the maximum possible error?

1
Rewrite the series in standard form: $a_{k} = \frac{1}{k ^{2}}$ , which is positive for all $k \geq 1$ .
2
Check the first condition:
3
$\lim_{k \to \infty} a_k = \lim_{k \to \infty} \frac{1}{k^2} = 0$
4
Check the second condition: $a_{k + 1} = \frac{1}{( k + 1 ) ^{2}} < \frac{1}{k ^{2}} = a_{k}$ for all $k \geq 1$ , so ${a_{k}}$ is strictly decreasing.
5
For $n = 4$ terms, the first neglected term is $a_{5} = \frac{1}{5 ^{2}} = 0.04$ . By the error bound, $∣ R_{4} ∣ \leq 0.04$ .
6
Conclusion: The error bound applies, maximum error is 0.04.

2. Constructing a Tight Interval for the Exact Sum★★★☆☆⏱ 4 min

Once you have the partial sum $S_{n}$ and the error bound, you can construct an interval guaranteed to contain the exact sum $S$ , a common AP exam question. In addition to the magnitude of the error, we know the error $R_{n}$ has the same sign as the first neglected term, which lets us create a tighter interval than the symmetric interval, which is what exam questions almost always expect.

If first neglected term is positive: $S_{n} < S \leq S_{n} + a_{n + 1}$
If first neglected term is negative: $S_{n} - a_{n + 1} \leq S < S_{n}$

📐 Worked Example

For the series $\sum_{k = 1}^{\infty} \frac{( - 1 ) ^{k + 1}}{2 ^{k} k}$ , find the tightest possible interval that contains the exact sum $S$ using the first 3 terms.

1
Calculate the 3rd partial sum:
2
$S_3 = \frac{1}{2^1 1} - \frac{1}{2^2 2} + \frac{1}{2^3 3} = \frac{10}{24} \approx 0.4167$
3
Confirm error bound conditions: $a_{k} = \frac{1}{2 ^{k} k}$ is positive, strictly decreasing, and $lim_{k \to \infty} a_{k} = 0$ , so the bound applies. The first neglected term is $a_{4} = \frac{1}{2 ^{4} 4} = 0.015625$ .
4
The 4th term of the series is $(- 1)^{4 + 1} a_{4} = - a_{4}$ , so it is negative, meaning $R_{3} = S - S_{3}$ is also negative, so $S < S_{3}$ .
5
Combine magnitude and sign: $- a_{4} \leq R_{3} < 0$ , so adding $S_{3}$ to all parts gives $S_{3} - a_{4} < S < S_{3}$ .
6
Substitute values to get the final interval:
7
$0.4167 - 0.015625 < S < 0.4167 \implies 0.401 < S < 0.417$

3. Finding Minimum Number of Terms for a Given Error Tolerance★★★☆☆⏱ 4 min

A common AP exam problem asks for the minimum number of terms needed to approximate the sum of an alternating series to within a given maximum error $E$ . By the error bound, $∣ R_{n} ∣ \leq a_{n + 1}$ , so we just need to find the smallest integer $n$ such that $a_{n + 1} < E$ . The most common mistake on this problem is confusing $n$ (number of terms used) with $n + 1$ (index of the first neglected term).

📐 Worked Example

How many terms of the alternating series $\sum_{k = 1}^{\infty} \frac{( - 1 ) ^{k + 1}}{k !}$ are needed to approximate the sum to within 0.001?

1
Confirm conditions: $a_{k} = \frac{1}{k !}$ is positive, strictly decreasing, and $lim_{k \to \infty} a_{k} = 0$ , so the error bound applies. We need $∣ R_{n} ∣ < 0.001$ , which requires $a_{n + 1} < 0.001$ .
2
Set up the inequality:
3
$\frac{1}{(n+1)!} < 0.001 \implies (n+1)! > 1000$
4
Calculate factorials: $6! = 720 < 1000$ and $7! = 5040 > 1000$ , so $n + 1 = 7$ .
5
Solve for $n$ : $n = 7 - 1 = 6$ . Verify: $a_{7} \approx 0.0002 < 0.001$ (meets requirement), while $n = 5$ gives $a_{6} \approx 0.0014 > 0.001$ (does not meet requirement).

✓ Quick check

Test your understanding with this AP-style multiple choice question:

How many terms of the convergent alternating series $\sum_{k = 1}^{\infty} \frac{( - 1 ) ^{k + 1}}{k ^{3}}$ are needed to approximate the sum to within 0.01 of the exact value?
- A) 3
- B) 4
- C) 5
- D) 10
Reveal answer
B) 4 —
Correct! We need $a_{n + 1} < 0.01$ , which simplifies to $(n + 1)^{3} > 100$ . The smallest integer satisfying this is $n + 1 = 5$ , so $n = 4$ terms are needed.

4. Common Pitfalls

Wrong move:

Applying the alternating series error bound to an alternating series where ${a_{k}}$ is not decreasing for all $k \geq n$ .

Why:

Students forget the decreasing condition applies to all terms after $n$ , not just the series as a whole.

Correct move:

Always confirm $a_{k + 1} < a_{k}$ for all $k \geq n$ before applying the bound, and only use the bound for $n \geq N$ , where $N$ is the first index where the sequence becomes decreasing.

Wrong move:

For a minimum terms problem, solving $a_{n} < E$ instead of $a_{n + 1} < E$ , leading to an answer one term too few.

Why:

Students confuse the index of the last term used with the index of the first neglected term.

Correct move:

Explicitly label your terms: if the last term you use is term $n$ , the first neglected term is term $n + 1$ , so the inequality is always $a_{n + 1} < E$ .

Wrong move:

Constructing a symmetric interval $S_{n} - a_{n + 1} < S < S_{n} + a_{n + 1}$ when asked for the tightest possible interval.

Why:

Students memorize the magnitude bound but forget the sign rule that gives a tighter interval.

Correct move:

Always check the sign of the first neglected term and adjust the interval to be one-sided, as shown in this guide.

Wrong move:

Applying the alternating series error bound to a divergent alternating series.

Why:

Students assume all alternating series are convergent.

Correct move:

Always confirm $lim_{k \to \infty} a_{k} = 0$ (the first AST condition) before applying the error bound, since the theorem only applies to convergent series.

Wrong move:

Using the alternating series error bound for the remainder of a positive-term Taylor series.

Why:

Students confuse alternating series error bound with Lagrange error bound.

Correct move:

Only use alternating series error bound for convergent alternating series meeting the AST conditions; use Lagrange error bound for general Taylor series with non-alternating signs.

5. Quick Reference Cheatsheet

Category	Formula/Rule	Notes
Standard Form	$\sum_{k = 1}^{\infty} (- 1)^{k + 1} a_{k}, a_{k} > 0$	Any alternating series can be rewritten to fit
Pre-Conditions	$lim_{k \to \infty} a_{k} = 0$ 2. ${a_{k}}$ strictly decreasing for $k \geq n$	Both AST conditions required for the bound
Error Bound	$∣ R_{n} ∣ = ∣ S - S_{n} ∣ \leq a_{n + 1}$	$S$ = exact sum, $a_{n + 1}$ = first neglected term
Error Sign	$sign (R_{n}) = sign(first neglected term)$	Used to build a tighter interval for $S$
Tight Interval for $S$	Positive neglected term: $S_{n} < S \leq S_{n} + a_{n + 1}$ Negative neglected term: $S_{n} - a_{n + 1} \leq S < S_{n}$	This is the expected answer for AP FRQs
Minimum Terms for Error $E$	Find smallest $n$ such that $a_{n + 1} < E$	$n$ = number of terms; common mistake uses $a_{n} < E$
Symmetric Interval	$S_{n} - a_{n + 1} < S < S_{n} + a_{n + 1}$	Always correct but wider than the tight interval

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · BC
FRQ error bound for alternating Taylor series
2019 · BC MCQ
Find minimum number of terms for error

What's Next

Alternating series error bound is a critical prerequisite for approximating sums of convergent series, and specifically for error analysis of alternating Taylor series, a commonly tested topic on the AP Calculus BC exam. After mastering this topic, you will apply alternating series error bound to find the error in alternating Taylor polynomial approximations; for these series, it is much simpler to apply than the Lagrange error bound, and it is a frequent source of points on both MCQ and FRQ sections. Without a solid understanding of this topic, you will struggle to justify error bounds for Taylor approximations, a high-weight FRQ skill, which makes up part of the ~17% of the AP Calculus BC exam devoted to infinite sequences and series.