Absolutely convergent series

A series of real or complex numbers, or more in general of elements in a Banach space $A$, \begin{equation}\label{e:serie} \sum_{i=0}^\infty a_n \end{equation} for which the series of real numbers \begin{equation}\label{e:serie_norme} \sum_{i=0}^\infty |a_n| \end{equation} converges (in the case of a general Banach space the series \ref{e:serie_norme} must be substituted by $\sum \|a_n\|_A$, where $\|\cdot\|_A$ denotes the norm in $A$). The absolute convergence implies the convergence. However the opposite is false, see for instance \[ \sum_{n=0}^\infty (-1)^n \frac{1}{n+1}\, . \]

A necessary and sufficient condition for the absolute convergence of a series is the Cauchy's criterion: for each $\varepsilon >0$ there exists $N_0$ such that \[ \sum_{n=k}^j |a_n| < \infty \qquad \left(\mbox{resp. } \sum_{n=k}^j \|a_n\|_A < \infty \mbox{ in a Banach space '"`UNIQ-MathJax7-QINU`"'}\right) \] for every $k>j>N$.

If a series converge absolutely any reordering gives also an absolutely convergent series and their limits coincide.

Linear combinations of absolutely convergent series are also absolutely convergent and their limits are the linear combinations of the original series. Cauchy products of absolutely convergent series are also absolutely convergent: in fact if $\sum a_n$ and $\sum b_n$ are absolutely convergent, then any series including all possible products $a_n b_m$ arranged in any order is also absolutely convergent. The corresponding limit is the product of the limits of the two series $\sum a_n$ and $\sum b_n$. This property remains true in a Banach algebra.

These properties of absolutely-convergent series are also displayed by multiple series: \begin{equation}\label{e:multiple} \sum_{n_1, \ldots n_k} a_{n_1 \ldots n_k} \end{equation} If a multiple series is absolutely convergent, it is convergent, for example, both in the sense of spherical and of rectangular partial sums, and its sums will be the same in both cases. If the multiple series (4) is absolutely convergent, the iterated series \begin{equation}\label{e:iterated} \sum_{n_1=0}^\infty \ldots \sum_{n_k=0}^\infty a_{n_1\ldots n_k} \end{equation} is absolutely convergent, i.e. all series obtained by successive summation of terms of the series \ref{e:multiple} by the indices $n_1, \ldots n_k$ are absolutely convergent; moreover, the sums of the multiple series \ref{e:multiple} and the iterated series \ref{e:iterated} are identical with the sum of any simple series formed by all terms of the series \ref{e:multiple}.

References

[AbSe1]	T.M. Apostol, "Mathematical analysis" , Addison-Wesley (1969)
[AbSe2]	V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)
[AbSe3]	L.D. Kudryavtsev, "Mathematical analysis" , 1 , Moscow (1973) (In Russian)
[Abse4]	S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian)