# Abel's Lemma

## Lemma

Let $\left \langle {a} \right \rangle$ and $\left \langle {b} \right \rangle$ be sequences in an arbitrary ring $R$.

### Formulation 1

- $\displaystyle \sum_{k \mathop = m}^n a_k \left({b_{k + 1} - b_k}\right) = a_{n + 1} b_{n + 1} - a_m b_m - \sum_{k \mathop = m}^n \left({a_{k + 1} - a_k}\right) b_{k + 1}$

### Formulation 2

Let $\displaystyle A_n = \sum_{i \mathop = m}^n {a_i}$ be the partial sum of $\left \langle {a} \right \rangle$ from $m$ to $n$.

Then:

- $\displaystyle \sum_{k \mathop = m}^n a_k b_k = \sum_{k \mathop = m}^{n - 1} A_k \left({b_k - b_{k + 1} }\right) + A_n b_n$

## Also known as

**Abel's lemma** is also known as **Abel's transformation**.

Some sources refer to it as the **technique of Summation by Parts**.

## Source of Name

This entry was named for Niels Henrik Abel.