Abel's Lemma/Formulation 1

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Lemma

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


Then:

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


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

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

$\blacksquare$


Also reported as

Some sources give this as:

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

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


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

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


Source of Name

This entry was named for Niels Henrik Abel.