Abel Test (series)

Abel Test: Let $\sum_{n=1}^\infty a_n$ be a convergent series of real numbers, and let $(b_n)$ be a bounded monotone sequence. Then the series $\sum_{n=1}^\infty a_n b_n$ converges.

Abel’s test is closely related to Dirichlet's test and is often used for power series at boundary points or for series with slowly varying factors.

Proof sketch (optional): Since $(b_n)$ is bounded and monotone, the differences $b_n-b_{n+1}$ have a fixed sign and sum absolutely. Combine summation by parts with convergence (hence bounded partial sums ) of $\sum a_n$.