Alternating series test

The alternating series test guarantees convergence when the positive term magnitudes of an alternating series decrease monotonically to zero.

Alternating series: An infinite series whose terms alternate in sign. The test applies to this family of series, typically written with positive magnitudes and alternating signs.
Infinite series: The sum of the terms of an infinite sequence, defined through the limit of its partial sums. The test determines convergence of this limiting sum, not merely behavior of individual terms.
Ratio test: A convergence test that uses the limiting ratio of successive term magnitudes. It can decide some series for which the alternating series test's monotonicity conditions are unavailable.
Alternating harmonic series: The series 1 − 1/2 + 1/3 − 1/4 + …, which converges conditionally. Its decreasing terms tend to zero, so it is a standard direct application of the test.
Monotone sequence: A sequence that never reverses direction, either increasing or decreasing. The test requires the magnitudes of successive terms to decrease monotonically.
Partial sum: The finite sum of the first specified terms of a sequence or series. Convergence in the test means these sums approach a finite limit.
Root test: A convergence test based on the limiting nth root of the absolute values of series terms. It analyzes term magnitudes directly rather than relying on alternating signs.
Leibniz formula for pi: The identity π/4 = 1 − 1/3 + 1/5 − 1/7 + …. The decreasing odd-reciprocal terms satisfy the test, though convergence is slow.
Limit of a sequence: A value that sequence terms approach as their indices grow without bound. The term magnitudes must approach zero for the alternating partial sums to converge.
Sequence convergence: The property of a sequence whose terms approach a finite limit as their indices increase. The alternating series converges when its sequence of partial sums converges.

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