Absolute convergence

In Mathematical, one says that a numerical series $\ sum u_n$ converges absolutely when the series of the absolute values (or the module S) $\ sum |u_n|$ is convergent. It is a very useful condition sufficient of convergence for the series $\ sum u_n$ itself. This sufficient condition can be wide with the series with values in a complete vector space normalized.

In a symmetrical way, it is said that a Intégrale converges absolutely if the integral of the absolute Value (of the module or the standard) of the intégrande is convergent.

The absolute convergence of the series or the integrals is closely related to the sommability ( families or functions), and brings properties stronger than convergence.

Absolutely convergent numerical series

A series with real or complex terms $\ sum a_n$ converges absolutely when the series of general term $| a_n |$ converges. In this case, the series $\ sum a_n$ also converges it and the triangular Inégalité spreads in

\ left|\ sum_ {n=0} ^ {+ \ infty} a_n \ right|\ Leq \ sum_ {n=0} ^ {+ \ infty} |a_n|

If the series is convergent, but not absolutely convergent, it is known as semi-convergent .

Example: the alternate harmonic series $\ sum_ {N \ Ge 1} \ frac {(- 1) ^n} {N}$ is semi-convergent.

Behavior of the series with real terms

If one deals with series of realities, the preceding theorem has an elementary demonstration, which brings extra informations on the possible behaviors.

If the $a_n$ terms of the series are realities, one can separate the positive and negative terms. It is necessary to consider for that the terms $a_n^+$ positive Partie and $a_n^-$ left negative the term $a_n$

a_n^+= \ max (a_n, 0) \ qquad a_n^-= \ max (- a_n, 0)

These two terms are positive, one is null, and the other equal one to the absolute value of

a_n

. So that

a_n = a_n^+-a_n^- \ qquad  |a_n| = a_n^++a_n^-

The series $\ sum a_n^+$ and $\ sum a_n^-$ being with positive terms, their continuation of the partial sums is increasing; it converges or tends towards the infinite one. Absolute convergence and semi-convergence can be formulated using these two series.

When the series $\ sum a_n$ converges absolutely, by comparison of positive series, the series $\ sum a_n^+$ and $\ sum a_n^-$ converge both, therefore by linearity the series $\ sum a_n$ too.
When the series $\ sum a_n$ is semi-convergent, necessarily the two series $\ sum a_n^+$ and $\ sum a_n^-$ diverge (each one has an infinite sum). Convergence is thus done by compensation between the positive and negative terms.

The property “absolute convergence implies convergence” can then be wide with the series with complex values by separating parts in the same way real and imaginary.

Properties of the absolutely convergent series

If a series is absolutely convergent, she enjoys properties particular, valid in general for the finished, but false sums for the series

generalization of the Commutation: the convergence and the value of the sum do not depend about the terms. Thus, if σ is a Permutation of ${\ mathbb NR}$ , it is possible to write

\ sum_ {n=0} ^ {+ \ infty} a_ {\ sigma (N)}= \ sum_ {n=0} ^ {+ \ infty} a_n

It should be noted that if the series is on the contrary semi-convergent a theorem of Riemann watch that to change the order of the terms can lead to a divergent series, or with a convergent series of arbitrarily selected nap.

generalization of the Distributivité: the Produit of Cauchy of two absolutely convergent series converges, with the formula

\ left (\ sum_ {p=0} ^ {+ \ infty} a_p \ right) \ left (\ sum_ {q=0} ^ {+ \ infty} b_q \ right) = \ sum_ {s=0} ^ {+ \ infty} \ left (\ sum_ {n=0} ^ S. has. _nb_ {Sn} \ right)

Another way of obtaining these properties for infinite sums is to consider the concept of summable Famille, very close to the property of absolute convergence for the numerical series.

Extension to the series with vectorial values

The framework is this time a vector Space normalized E . A series with vectorial terms $\ sum a_n$ converges absolutely when the series of general term $\| a_n \|$ converges.

When the vector space E is complete, absolute convergence still provides a sufficient condition of convergence: if the series converges absolutely, it converges and

\ left \|\ sum_ {n=0} ^ {+ \ infty} a_n \ right \|\ Leq \ sum_ {n=0} ^ {+ \ infty} \|a_n \|

This property is proven by using the Critère of Cauchy to characterize these convergences.

It is acted in fact of an equivalence: if E is a vector space normalized such as any absolutely convergent series converges, then E is complete.

Absolutely convergent integral

In the same way, an integral:

\ int_A F (X) \, dx

converge absolutely if the integral of its corresponding absolute value is finished:

\ int_A \ left|F (X) \ right|\, dx< \ infty.

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