Dirichlet Test (series)

Dirichlet Test: Let $(a_n)$ and $(b_n)$ be real sequences such that:

• the partial sums $A_N=\sum_{n=1}^N a_n$ are bounded, and
• $b_n$ is monotone with $b_n\to 0$. Then the series $\sum_{n=1}^\infty a_n b_n$ converges .

Dirichlet’s test is a powerful tool for oscillatory series where cancellation occurs through bounded partial sums.

Proof sketch (optional): Use summation by parts: $\sum_{n=1}^N a_n b_n = A_N b_{N+1} + \sum_{n=1}^N A_n (b_n-b_{n+1}),$ and show the right-hand side is Cauchy using boundedness of $A_n$ and the telescoping nature of $b_n-b_{n+1}$.