Abel summation
Integration by parts is the move that trades a hard integral of a product for an easier one. Abel summation is its discrete twin: a way of re-bundling a sum of products sum a_n b_n so that, instead of the original terms, you work with partial sums and differences. It is the algebraic engine behind many convergence tests for series.
The identity (summation by parts): let A_n = a_1 + a_2 + ... + a_n be the partial sums of (a_n). Then sum_{n=1}^N a_n b_n = A_N b_N - sum_{n=1}^{N-1} A_n (b_{n+1} - b_n). The product a_n b_n has been re-expressed using the partial sums A_n and the consecutive differences of b_n, exactly mirroring the integral of u dv becoming uv minus the integral of v du.
Its power shows when one factor has bounded partial sums and the other decreases to zero. Then A_N b_N stays controlled and the differences telescope nicely, forcing convergence — this is the mechanism behind Dirichlet's test and Abel's test, which generalize the alternating series test to series like sum (cos n x)/n. Abel summation is purely algebraic, so it needs no continuity or differentiability; it is a tool for taming series of products.
To show sum sin(n)/n converges: the partial sums of sin(n) are bounded (a trigonometric identity caps them), while 1/n decreases to 0. Abel summation then makes A_N (1/N) -> 0 and the remaining differences sum absolutely, giving convergence.
Convergence of sum sin(n)/n by summation by parts.