Summation by parts

In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826. Suppose and are two sequences. Then, Using the forward difference operator , it can be stated more succinctly as Summation by parts is an analogue to integration by parts: or to Abel's summation formula: An alternative statement is which is analogous to the integration by parts formula for semimartingales. Although applications almost always deal with convergence of sequences, the statement is purely algebraic and will work in any field. It will also work when one sequence is in a vector space, and the other is in the relevant field of scalars. The formula is sometimes given in one of these - slightly different - forms which represent a special case () of the more general rule both result from iterated application of the initial formula. The auxiliary quantities are Newton series: and A particular () result is the identity Here, is the binomial coefficient. For two given sequences and , with , one wants to study the sum of the following series: If we define then for every and Finally This process, called an Abel transformation, can be used to prove several criteria of convergence for . The formula for an integration by parts is . Beside the boundary conditions, we notice that the first integral contains two multiplied functions, one which is integrated in the final integral ( becomes ) and one which is differentiated ( becomes ). The process of the Abel transformation is similar, since one of the two initial sequences is summed ( becomes ) and the other one is differenced ( becomes ). It is used to prove Kronecker's lemma, which in turn, is used to prove a version of the strong law of large numbers under variance constraints. It may be used to prove Nicomachus's theorem that the sum of the first cubes equals the square of the sum of the first positive integers.