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Abel test

A convergence test for sums of products when one series converges and the other factor is monotone and bounded.

Abel test: Consider a series ∑n=1∞anbn\sum_{n=1}^\infty a_n b_n of real or complex numbers. Suppose

  1. the series ∑n=1∞an\sum_{n=1}^\infty a_n , and
  2. the sequence (bn)(b_n) is monotone and bounded.

Then the series ∑n=1∞anbn\sum_{n=1}^\infty a_n b_n converges.

Remarks

Abel’s test can be viewed as complementary to the : Dirichlet assumes bounded of (an)(a_n) and bn→0b_n\to 0, while Abel assumes convergence of ∑an\sum a_n and only boundedness of (bn)(b_n).