Abel’s Test

Abel’s test, in analysis (a branch of mathematics), a test for determining if an infinite series converges to some finite value. The test is named for the Norwegian mathematician Niels Henrik Abel (1802–29).

Starting with any known convergent series, say Σ a_n (i.e., a₁ + a₂ + a₃ + ⋯), Abel proved that, for a sequence of monotonically decreasing positive numbers b_n (i.e., b₁ ≥ b₂ ≥ b₃ ≥ ⋯ > 0), the infinite series Σ a_nb_n (a₁b₁ + a₂b₂ + a₃b₃ + ⋯) converges to some finite value. In practice, to use Abel’s test one begins with an infinite series and factors each term in the sequence in such a way that one of the factors produces a known convergent series and the other factor produces a monotonically decreasing sequence of positive numbers.