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Dirichlet test

A convergence test for sums of products using bounded partial sums and monotone factors.

Dirichlet test: Consider a series ∑n=1∞anbn\sum_{n=1}^\infty a_n b_n, where ana_n may be real or complex and bnb_n is real. Let

An=∑k=1nakA_n=\sum_{k=1}^n a_k

denote the of ∑an\sum a_n. If

  1. the sequence (An)(A_n) is bounded, and
  2. (bn)(b_n) is monotone and bn→0b_n\to 0,

then ∑n=1∞anbn\sum_{n=1}^\infty a_n b_n .

Remarks

The is a special case (take an=(−1)n−1a_n=(-1)^{n-1}), and Dirichlet’s test is closely paired with the .