The Alternating Series Error Bound

A partial sum is only an estimate of the sum of a series. For most series it's hard to say how good the estimate is. For an alternating series that passes the alternating series test, the answer comes straight from the terms.

Take $\sum (-1)^{n+1} b_n$ with $b_n > 0$ decreasing to 0. Its partial sums step back and forth past the sum $S$, each step shorter than the one before. So $S$ always lies between two consecutive partial sums $S_n$ and $S_{n+1}$, which differ by $b_{n+1}$.

If the first omitted term is positive, adding it raises the estimate, so $S_n$ is too small. If it's negative, $S_n$ is too large. When the terms decrease only from some $N$ on, the bound holds for $S_n$ with $n \ge N$.

Run the bound backward to find how many terms make the error small enough. Find the first $n$ with $b_{n+1}$ below the target.