Concept preview

What you will be able to do

1. 01State the alternating series error bound: for a convergent alternating series, the error $|S-S_N|\le b_{N+1}$, the first OMITTED term.
2. 02Use the bound to estimate the accuracy of a partial sum $S_N$ and to state a guaranteed interval containing the true sum $S$.
3. 03Determine how many terms are needed to approximate the sum of an alternating series to within a specified tolerance, correctly using $b_{N+1}$ rather than $b_N$.

Let $\sum(-1)^{n-1}b_n$ (with each $b_n>0$) be a series that satisfies the alternating series test: the terms $b_n$ decrease monotonically and $b_n\to 0$. Write $S$ for the true sum and $S_N=\sum_{n=1}^{N}(-1)^{n-1}b_n$ for the $N$th partial sum. The alternating series error bound states that the remainder is no larger than the first term you dropped: $|S-S_N|\le b_{N+1}.$ In words: if you stop after $N$ terms, the error is at most the size of term number $N+1$ — the FIRST OMITTED term. This is one of the sharpest, simplest error statements in all of calculus.