Alternating-series error bounds and approximation intervals

Understand the idea

For an alternating series whose positive magnitudes decrease to zero, the error after a partial sum is no larger than the magnitude of the first omitted term. The sign of that omitted term also indicates which side of the true sum the partial sum lies on.

Identify the last included index and the first omitted index explicitly. The error bound uses the omitted term, not the last term that was added. To meet a tolerance, solve the corresponding inequality for a whole-number index and respect whether the tolerance is strict.

The error bound is a guarantee, not the exact error. Consecutive partial sums bracket the limit under the alternating-series conditions. This gives an interval containing the sum without requiring a decimal approximation of the exact value.

Worked example

Use four terms to approximate $\displaystyle \ln2=\sum_{n=1}^{\infty}(-1)^{n-1}/n$, and give a guaranteed interval.

1. The fourth partial sum is $\displaystyle S_4=1-1/2+1/3-1/4=7/12$.
2. The next term is positive and has magnitude $\displaystyle 1/5$.
3. Therefore the true sum lies above $\displaystyle S_4$ and below $\displaystyle S_5=7/12+1/5=47/60$.