Topic 10.10 Alternating Series Error Bound: bound the error of a partial-sum approximation of an alternating series by the magnitude of the first omitted term (BC).

What this topic is asking

The College Board (Topic 10.10, BC only) gives the alternating series error bound, a remarkably simple way to control the error when you approximate the sum of a convergent alternating series by stopping after finitely many terms. The error is no larger than the first term you left out.

The bound

Why the next term bounds the error

The bound comes straight from the oscillation that makes alternating series converge (Topic 10.7). Because the partial sums overshoot and undershoot the true sum $S$ in ever-smaller steps, the true sum is always trapped between two consecutive partial sums $S_n$ and $S_{n+1}$. The gap between those partial sums is exactly the next term $b_{n+1}$, so $S$ sits within $b_{n+1}$ of $S_n$. That is the whole proof in one sentence: the limit lies between consecutive partial sums, which differ by $b_{n+1}$. This geometric trapping is why no derivative bounds or factorials are needed, unlike the general Taylor error bound.

A worked error estimate

Choosing how many terms for a given accuracy

A common reverse question asks how many terms you need so the approximation is accurate to within a stated tolerance $\varepsilon$. Since the error after $n$ terms is at most $b_{n+1}$, you require $b_{n+1} \le \varepsilon$ and solve for the smallest such $n$. For $\sum\frac{(-1)^{n+1}}{n}$ to be accurate within $0.01$, you need $\frac{1}{n+1} \le 0.01$, so $n + 1 \ge 100$, meaning $n \ge 99$ terms. This direction is just the bound rearranged, and it is a standard free-response setup. The work expected is the inequality $b_{n+1} \le \varepsilon$ and the resulting value of $n$.

Limits of the bound

The alternating series error bound is only valid when the series genuinely alternates and satisfies the test (decreasing magnitudes tending to zero). For a non-alternating series, or for the error of a general Taylor polynomial whose series is not alternating at the point in question, you must use the Lagrange error bound (Topic 10.12) instead. Many Taylor approximations do happen to be alternating (for example $\sin x$, $\cos x$, and $\ln(1+x)$ at suitable points), and there the simpler alternating bound applies, which is why these often appear together. Recognizing whether the series you are truncating is alternating decides which error tool to reach for.