# Abel's Lemma/Formulation 1

## Lemma

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

Then:

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

Note that although proved for the general ring, this result is usually applied to one of the conventional number fields $\Z, \Q, \R$ and $\C$.

### Corollary

$\displaystyle \sum_{k \mathop = 1}^n a_k \left({b_{k + 1} - b_k}\right) = a_{n + 1} b_{n + 1} - a_1 b_1 - \sum_{k \mathop = 1}^n \left({a_{k + 1} - a_k}\right) b_{k + 1}$

## Proof

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

$\blacksquare$

## Also reported as

Some sources give this as:

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

which is obtained from the main result by interchanging $a$ and $b$.

Others take the upper index to $n - 1$:

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

## 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.