Alternating Series Error Bound BC only

Because consecutive partial sums bracket the limit of a convergent alternating series, the distance from $S_n$ to $S$ is at most the distance from $S_n$ to $S_{n+1}$, which is a single term, giving $|S - S_n| \le b_{n+1}$. The bound is therefore the first term omitted rather than the last one used, and the off-by-one survives because it errs in the safe direction, since $b_n \ge b_{n+1}$ makes the last term used a valid if weaker bound. At $n = 4$ for the alternating harmonic series the actual error is $0.1098$ and the bound is $b_5 = 0.2$, available without knowing the sum is $\ln 2$.

The off-by-one becomes costly on the standard how-many-terms question: making the error less than $0.01$ needs $\frac{1}{n+1} < 0.01$, so $n \ge 100$, while solving $\frac{1}{n} < 0.01$ gives $101$. Distinguishing this bound from Lagrange's is a matter of machinery, since this one is simply a term of the series while Lagrange needs a derivative bounded over an interval, a factorial and a power of $(x-a)$; where both apply, as with a Maclaurin series that alternates at the point in question, the alternating bound is far easier. It also requires the same two conditions as the convergence test, because it is a consequence of the bracketing rather than a separate fact, so a failure of the decrease condition makes the bound false rather than merely unavailable.