In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, summation by parts transforms the summation

Summation

Summation is the operation of adding a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation. The numbers to be summed may be integers, rational numbers,...

 of products of sequences

Séquences

Séquences is a French-language film magazine originally published in Montreal, Quebec by the Commission des ciné-clubs du Centre catholique du cinéma de Montréal, a Roman Catholic film society. Founded in 1955, the publication was edited for forty years by Léo Bonneville, a member of the Clerics...

 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

Niels Henrik Abel

Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...

 lemma or Abel transformation.

Statement

Suppose and are two sequence

Sequence

In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

s. 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

Integration by parts

In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...

 formula,

Note also that although applications almost always deal with convergence of sequences, the statement is purely algebraic and will work in any field

Field

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. It will also work when one sequence is in a vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

, and the other is in the relevant field of scalars.

Newton series

The formula is sometimes given in one of these - slightly different - forms


which represent a special cases () of the more general rule


both result from iterated application of the initial formula. The auxiliary quantities are Newton series:

and

A remarkable, particular () result is the noteworthy identity

Here, is the binomial coefficient

Binomial coefficient

In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...

.

Method

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 .

Similarity with an integration by parts

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 ).

Applications

We suppose that ; otherwise it is obvious that is a divergent series

Divergent series

In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a limit....

.

If is bounded by a real M and is absolutely convergent, then is a convergent series.

And the sum of the series verifies:

See also

• Convergent series
• Divergent series
  Divergent series
  In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a limit....
  
  
• Integration by parts
  Integration by parts
  In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...
  
  
• Cesàro summation
  Cesàro summation
  In mathematical analysis, Cesàro summation is an alternative means of assigning a sum to an infinite series. If the series converges in the usual sense to a sum A, then the series is also Cesàro summable and has Cesàro sum A...
  
  
• Abel's theorem
  Abel's theorem
  In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.-Theorem:...
  
  
• Abel sum formula