alternating series test

When the terms of a series flip sign at every step — plus, minus, plus, minus — the partial sums no longer climb in one direction but jump back and forth, each overshoot smaller than the last. If the sizes of the terms steadily shrink to zero, those oscillations close in on a single value, like a damped pendulum settling. That is the alternating series test.

Precisely (Leibniz): let b_n > 0 with b_n decreasing (b_{n+1} <= b_n) and b_n -> 0. Then the alternating series sum (-1)^{n} b_n (or sum (-1)^{n+1} b_n) converges. Moreover the sum lies between any two consecutive partial sums, which yields the very useful error bound: the remainder after N terms is at most b_{N+1} in absolute value, and has the sign of the first omitted term.

Both hypotheses matter. Monotone decrease cannot be dropped without care — if the b_n merely tend to zero but not monotonically, the conclusion can fail. And the test gives only convergence, not absolute convergence; a series may pass this test yet have its absolute-value series diverge, in which case the convergence is merely conditional.