# Abel's Lemma

## Lemma

Let $\sequence a$ and $\sequence b$ be sequences in an arbitrary ring $R$.

### Formulation 1

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

### Formulation 2

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

Then:

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

## Also known as

Abel's Lemma is also known as:

Abel's transformation
Abel's partial summation formula
the technique of Summation by Parts.

## Source of Name

This entry was named for Niels Henrik Abel.