Abel's Lemma/Formulation 1

Lemma

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

Then:

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

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

$\ds \sum_{k \mathop = 1}^n a_k \paren {b_{k + 1} - b_k} = a_{n + 1} b_{n + 1} - a_1 b_1 - \sum_{k \mathop = 1}^n \paren {a_{k + 1} - a_k} b_{k + 1}$

Proof

\(\ds \sum_{k \mathop = m}^n a_k \paren {b_{k + 1} - b_k}\)

\(=\)

\(\ds \sum_{k \mathop = m}^n a_k b_{k + 1} - \sum_{k \mathop = m}^n a_k b_k\)

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = m}^n a_k b_{k + 1} - \paren {a_m b_m + \sum_{k \mathop = m}^n a_{k + 1} b_{k + 1} - a_{n + 1} b_{n + 1} }\)

\(\ds \)

\(=\)

\(\ds a_{n + 1} b_{n + 1} - a_m b_m + \sum_{k \mathop = m}^n a_k b_{k + 1} - \sum_{k \mathop = m}^n a_{k + 1} b_{k + 1}\)

\(\ds \)

\(=\)

\(\ds a_{n + 1} b_{n + 1} - a_m b_m - \sum_{k \mathop = m}^n \paren {a_{k + 1} - a_k} b_{k + 1}\)

$\blacksquare$

Also reported as

Some sources give this as:

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

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

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

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

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.