Summation by parts

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"Abel transformation" redirects here. For another transformation, see Abel transform.

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. The summation by parts formula is sometimes called Abel's lemma or Abel transformation.


Suppose and are two sequences. Then,

Using the forward difference operator , it can be stated more succinctly as

Note that summation by parts is an analogue to the integration by parts formula,

Note also that 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.

Newton series[edit]

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:


A remarkable, particular () result is the noteworthy 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


This process, called an Abel transformation, can be used to prove several criteria of convergence for .

Similarity with an integration by parts[edit]

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 ).


The Cauchy criterion gives

where a is the limit of . As is convergent, is bounded independently of , say by . As go to zero, so go the first two terms. The third term goes to zero by the Cauchy criterion for . The remaining sum is bounded by

by the monotonicity of , and also goes to zero as .

  • Using the same proof as above, one shows that
  1. if the partial sums form a bounded sequence independently of  ;
  2. if (so that the sum goes to zero as goes to infinity) ; and
  3. if

then is a convergent series.

In both cases, the sum of the series satisfies:

See also[edit]