Abel's Lemma

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Lemma

Let $\sequence a$ and $\sequence b$ 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 $\sequence a$ 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.