Abel Summation

Summation by parts: trade a sum of products for differences of one factor.

The idea

Theorem (Abel summation).

Let $a_1, \dots, a_n$ and $b_1, \dots, b_n$ be real numbers, and write $A_k = a_1 + a_2 + \cdots + a_k$ for the partial sums of the first sequence, with $A_0 = 0$. Then $\sum_{k=1}^{n} a_k b_k = A_n b_n - \sum_{k=1}^{n-1} A_k\,(b_{k+1} - b_k).$

The identity is also called summation by parts. It is an identity, not an estimate: both sides are the same number for every choice of the two sequences. The right side is a boundary term $A_n b_n$ and a sum against the consecutive differences $b_{k+1} - b_k$. For positive increasing $b_k$ the identity is a picture: columns of width $a_k$ and height $b_k$, together with rows of width $A_k$ and height $b_{k+1} - b_k$, tile the $A_n \times b_n$ rectangle, so the two ways of counting its area are the two sides of the identity.

Abel summation is the discrete form of integration by parts, $\int u\,dv = uv - \int v\,du$: a boundary term minus a sum in which the differencing has moved from one factor to the other. The $a_k$ now enter only through their partial sums, and the $b_k$ only through their differences. This exchange is what makes the identity useful: a sum of products that resists term-by-term estimates becomes manageable when the partial sums $A_k$ are bounded and the differences of $b$ are controlled — when $b$ is monotone, say, or slowly varying.

Ways to work on it

• Walkthrough. The summation-by-parts identity and why it aggregates cancellation.
• Practice. Evaluate a sum two ways via the Abel-summation identity.
• Hardest. Bound an oscillating sum against a monotone weight.

Not sure where to start? Take the ten-question placement test.