Dirichlet's test

Statement

Let $(a_n)$ be a monotone real sequence with $a_n \to 0$. Let $(b_n)$ be a real sequence whose partial sums are bounded. That is, there is $M$ with $\left| \sum_{k=1}^{n} b_k \right| \le M$ for all $n$. Then the series $\sum_{n=1}^{\infty} a_n b_n$ converges.

Remarks

The proof uses summation by parts and the Cauchy criterion for series. With $b_n = (-1)^n$ it gives the alternating series test. It also shows that $\sum \sin(n x) / n$ converges for each real $x$. Mathlib states a form with a Cauchy sequence of partial sums, as Monotone.cauchySeq_series_mul_of_tendsto_zero_of_bounded.

History

The test is named after Peter Gustav Lejeune Dirichlet. It was published after his death, in 1862.

Explanations

L1 human

The partial sums of $\sum b_n$ do not grow, and the weights $a_n$ shrink steadily to $0$. Together these give enough cancellation for $\sum a_n b_n$ to converge.