Abel's Lemma

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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.