---
tag: 9G82Z17DKA
kind: statement
role: theorem
name: Summation by parts
msc: [39A12, 40A05]
uses: [TXTR68V4R1, 0GYGHKX834]
about: [FT9JQA58KJ]
sources:
  - url: https://en.wikipedia.org/wiki/Summation_by_parts
same_as:
  wikidata: Q318754
  wikipedia: Summation by parts
---

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]], let $(f_k)$ and $(g_k)$ be sequences in $R$, and let $m \le n$ be integers.
Write $\Delta f_k = f_{k+1} - f_k$ and $\Delta g_k = g_{k+1} - g_k$ for the [[forward differences|TXTR68V4R1]].
Then
$$\sum_{k=m}^{n} f_k\, \Delta g_k = f_{n+1} g_{n+1} - f_m g_m - \sum_{k=m}^{n} g_{k+1}\, \Delta f_k.$$

## Remarks

The product rule for differences is $\Delta(f g)_k = f_k \Delta g_k + g_{k+1} \Delta f_k$, which one checks by expanding.
Summing it from $k = m$ to $n$, the left side telescopes to $f_{n+1} g_{n+1} - f_m g_m$, and this gives the formula.
A common form, often called the Abel transformation, uses partial sums $A_k = a_1 + \dots + a_k$:
$$\sum_{k=1}^{n} a_k b_k = A_n b_n - \sum_{k=1}^{n-1} A_k (b_{k+1} - b_k).$$
Example: with $f_k = k$ and $g_k = 2^k$, $\Delta g_k = 2^k$ and the formula gives $\sum_{k=0}^{n} k\, 2^k = (n+1) 2^{n+1} - \sum_{k=0}^{n} 2^{k+1} = (n - 1) 2^{n+1} + 2$.
For $n = 2$ both sides equal $10$.

## Explanations

### L1 human

Summation by parts moves a difference from one factor of a product to the other, at the price of boundary terms.
It is the discrete form of [[integration by parts|08Y1BBQTK9]].
It helps when one factor is easy to sum and the other changes slowly.

### L2 human

The Abel form is the key step in [[Dirichlet's test|0V9ME8GQE7]] and [[Abel's test|0CN97GADE5]] for the convergence of series $\sum a_k b_k$.
In Dirichlet's test, bounded partial sums $A_k$ and monotone $b_k$ tending to $0$ make the right side converge.
Its continuous-variable version is [[Abel's summation formula|V5EPT2PX9Z]], used in analytic number theory.
In the [[calculus of finite differences|FT9JQA58KJ]] it is proved from the [[fundamental theorem of the calculus of finite differences|MNSSV4QHD1]] applied to the product $f g$.