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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Summation is the addition of a set of numbers; the result is their sum. ...

Definition

Suppose {f_k} and {g_k} are two sequences. Then, In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...

.

Using the forward difference operator Δ, it can be stated more succinctly as In mathematics, a difference operator maps a function f(x) to another function f(x + a) − f(x + b). ...

Note that summation by parts is an analogue to the integration by parts formula, In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...

Newton series

The formula is sometimes stated in the slightly different form

,

which itself is a special case (M = 1) of this more general rule

,

which results from iterated application of the initial formula. The auxiliary quantities are Newton series: In mathematics, a difference operator maps a function, f(x), to another function, f(x + a) − f(x + b). ...

and

.

Here, is the binomial coefficient. In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is the number of combinations that exist. ...

Method

For two given sequences and , with , one wants to study the sum of the following series:

If we define ,
then for every n>0,

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 derivated ( becomes ).

The process of the Abel transformation is similar, since one of the two initial sequences is summed ( becomes ) and the other one is discretely derivated ( becomes ).

Applications

Let's consider that , otherwise it is obvious that is a divergent series. In mathematics, a divergent series is an infinite series that does not converge. ...

If is bounded by a real M and is absolutely convergent, then is a convergent series. The term bounded appears in different parts of mathematics where a notion of size can be given. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, a series is the sum of the terms of a sequence of numbers. ...

And the sum of the series verifies:

See also

• Convergent series
• Divergent series
• Integration by parts
• Abel's theorem
• Abel transform

In mathematics, a series is the sum of the terms of a sequence of numbers. ... In mathematics, a divergent series is an infinite series that does not converge. ... In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ... In real analysis, Abels theorem for power series relates a limit of a power series to the sum of its coefficients. ... In mathematics, the Abel transform, named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions. ...

References

• Abel's lemma on PlanetMath