Calculus BC · Unit 10: Infinite Sequences and Series · 14 min read · Updated 2026-05-10
Alternating Series Error Bound — AP Calculus BC
AP Calculus BC · Unit 10: 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.
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.
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: $S_n - a_{n+1} \leq S < S_n$
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).
Common Pitfalls
Why: Students forget the decreasing condition applies to all terms after $n$, not just the series as a whole.
Why: Students confuse the index of the last term used with the index of the first neglected term.
Why: Students memorize the magnitude bound but forget the sign rule that gives a tighter interval.
Why: Students assume all alternating series are convergent.
Why: Students confuse alternating series error bound with Lagrange error bound.