Alternating Series Error Bound

Use the first omitted alternating magnitude to give a justified error bound and interval for the sum.

College Board Topic 10.10 · CED p. 190 (PDF p. 195) · CED effective Fall 2020; project course outline 2026–27.

If an alternating series satisfies the decreasing-to-zero magnitude conditions, and you truncate after N terms in that decreasing tail, then $| S - S_{N} | \leq b_{N + 1}$ . The next omitted magnitude bounds the error. It is not the last retained term, and it is not normally the exact error.

The sum lies between consecutive partial sums $S_{N}$ and $S_{N + 1}$ . The next term also tells which side of Sₙ contains the true sum: a positive next term puts the sum above Sₙ; a negative next term puts it below. Keep the first index and number of retained terms explicit.

To guarantee a requested tolerance, solve an inequality for the next omitted term. If a strict error less than ε is requested, choose a bound strictly less than ε unless you can separately justify strictness. Do not silently substitute an at-most claim for a less-than claim.

For the alternating harmonic series, $S_{4} = 7 / 12$ and the next term is 1/5. Thus the sum lies between 7/12 and 47/60, with absolute error at most 1/5. The approximation is crude, but its uncertainty is honestly quantified.

Let's work through it

Sum four alternating harmonic terms: $S_{4} = 1 - 1 / 2 + 1 / 3 - 1 / 4 = 7 / 12$ . Why: Use the actual first four terms in order.
The fifth term has magnitude $1 / 5$ . Why: This is the first omitted term.
$| S - S_{4} | \leq 1 / 5$ , with $7 / 12 \leq S \leq 47 / 60$ . Why: Decreasing magnitudes tending to zero make adjacent partial sums trap the total.

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Question 1 of 20 self-assessed

Question 1

After N terms, which magnitude bounds the error?

Your answer and reasoning

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Question 1: prompt and worked reasoning

After N terms, which magnitude bounds the error?

b_{N + 1}

The first omitted term separates adjacent bounding partial sums.

Question 2: prompt and worked reasoning

Is the alternating error bound the exact error?

Generally no.

It is a guaranteed upper bound, not an equality.

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Retrieval 1

After N terms, which magnitude bounds the error?

Worked answer and reason

b_{N + 1}

The first omitted term separates adjacent bounding partial sums.

Retrieval 2

Is the alternating error bound the exact error?

Worked answer and reason

Generally no.

It is a guaranteed upper bound, not an equality.

Catch the tempting mistake

Use the last included term as the sharp alternating bound.: Use the first omitted magnitude after verifying the tail conditions.

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Unit 10 problem 19 · No calculator · analytic
Unit 10 problem 20 · No calculator · analytic
Unit 10 problem 30 · Graphing calculator required · multiple

For the teacher

Ask for an interval around the approximation, not only a naked error number. Check index counting with a series that starts at n=0.

Unit planning and formative guidance →

Glossary and deeper reading · 3 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

Alternating series
An alternating series has successive terms that switch between positive and negative signs. The alternating series test establishes convergence when the term magnitudes eventually decrease and approach 0. For example, 1−1/2+1/3−1/4+⋯ satisfies those conditions. Alternating signs alone do not guarantee convergence.
Read more: 10.7 · Alternating series →
Partial sum
A partial sum adds the first finitely many terms of a series. The nth partial sum is Sₙ=a₁+a₂+⋯+aₙ when indexing begins at 1. For example, the first three terms of 1/2+1/4+1/8+⋯ give partial sum 7/8. Convergence of a series concerns the limit of these sums, not just the limit of the individual terms.
Read more: 10.1 · Partial sum →
Harmonic series
The harmonic series is 1+1/2+1/3+1/4+⋯. It diverges even though its individual terms approach 0. For example, grouping terms from 1/2 through 1/3+1/4 and then 1/5+⋯+1/8 shows repeatedly added blocks of at least 1/2. It is a standard comparison that separates a necessary term condition from actual convergence.
Read more: 10.5 · Harmonic series →

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Question 1

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Alternating series

Partial sum

Harmonic series

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