Alternating Series

An alternating series is simply one whose signs take turns: plus, minus, plus, minus. Adding it up is like taking a step forward, a smaller step back, a smaller step forward, and so on, and you end up homing in on one spot. It's good because cancellation can make a series converge that otherwise wouldn't, and because it hands you a free bound on your error. The condition $\sum (-1)^n b_n$ with $b_n > 0$, $b_{n+1} \le b_n$, $b_n \to 0$ looks like a list of rules, but each piece is simple: $(-1)^n$ (the signs take turns, plus then minus), $b_n$ (how big each step is), $b_{n+1} \le b_n$ (each step is no bigger than the last) and $b_n \to 0$ (the steps eventually get tiny). The error bound $|s - s_n| \le b_{n+1}$ just says you're never farther from the answer than the next step. It's three easy checks dressed up in symbols, so bear with it.

Every test so far has been about positive terms. The comparison and integral tests both lean on the running total only going up. Once the signs start flipping, that picture breaks, and something new can happen: terms can cancel each other out.

Why it converges. Step forward $b_1$, back $b_2$, forward $b_3$, and so on. Each step is shorter than the last, so you never get all the way back to where you were. That means the even partial sums creep up, the odd ones creep down, and they never cross. The gap between them is just the next step, which shrinks to 0, so they squeeze onto a single number. There's a video further down that shows this hopping.

Checking the conditions.

1. The terms really alternate in sign.
2. The sizes decrease, at least eventually. Check $b_{n+1} \le b_n$ directly or show $f'(x) < 0$.
3. The sizes go to 0. If they don't, the series diverges by the divergence test.

The alternating harmonic series $1 - \frac12 + \frac13 - \frac14 + \cdots$ converges (to $\ln 2$), even though $\sum \frac1n$ diverges. Think about how surprising that is. Same sizes, just with every other sign flipped, and the infinite total becomes a finite one. Cancellation matters.

A common mistake. It's tempting to think any series with alternating signs converges, because the plus and minus "balance out." They don't always. $\sum (-1)^n \frac{n}{n+1}$ alternates, but its steps head to size 1, not 0, so the partial sums keep hopping by about 1 forever. Check all three conditions, and especially check that the sizes go to 0.

Error bound. Since the true sum is always trapped between two consecutive partial sums, stopping after $n$ terms leaves an error of at most $b_{n+1}$, with the same sign as that first omitted term. You get a free accuracy guarantee, which hardly any other test gives you.

Alternating series give computers a built-in stopping rule: keep adding terms until the next one is smaller than the accuracy you need. Calculators evaluate $\sin x$ and $\cos x$ this way.