Alternating Series Test for Convergence

A series like 1 - 1/2 + 1/3 - 1/4 + ... keeps changing direction. Each partial sum jumps past the limit, then the next term pulls it back, but less far each time. That back-and-forth is exactly what lets a series converge even when the series of absolute values would not. Compare it to the plain harmonic series 1 + 1/2 + 1/3 + ...: that one diverges because every term pushes the sum higher, with nothing to cancel the buildup. Flip every other sign, and now the pushes cancel the pulls almost completely, and the leftover gets tinier each round.

The mechanism is a squeeze. Write the terms as (-1)ⁿ·b_n where b_n > 0. If b_n decreases and b_n → 0, the partial sums S₁, S₂, S₃, ... zigzag: odd partial sums only go down, even ones only go up, and the two sequences close in on each other. Since the gap between consecutive partial sums is b_(n+1), and that shrinks to 0, the sums are trapped between tightening bounds and must converge to a single limit squeezed between any two consecutive partial sums. This also hands you a free error bound: the truncation error after n terms is at most b_(n+1), the size of the first dropped term. That bound is why this test connects directly to the next topic, estimating sums and bounding error for alternating series, and it contrasts with absolute convergence, where no such easy error estimate exists.

The test only checks two things about b_n: does it tend to 0, and is it eventually decreasing. It says nothing about whether the series converges absolutely, so it sits apart from the ratio and comparison tests, which test absolute convergence instead.