Absolutely convergent series

2020 Mathematics Subject Classification: Primary: 40F05 [MSN][ZBL]

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 (see Section 3.45 of [Ru]). 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 (cp. with Theorem 3.22 of [Ru]): for each $\varepsilon >0$ there exists $N$ such that \[ \sum_{n=j}^k |a_n| < \varepsilon \qquad \left(\mbox{resp. } \sum_{n=j}^k \|a_n\|_A < \varepsilon \mbox{ in a Banach space } A \right) \] for every $k>j>N$.

If a series converges absolutely any reordering gives also an absolutely convergent series and their sums coincide (see Theorem 3.55 of [Ru]).

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 $\sum b_n$ (see Theorem 3.50 of [Ru]). 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 \ref{e:multiple} 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}.

Relations to Lebesgue integral

It is possible to treat absolutely convergent series as a special case of Lebesgue integrals. To this end the countable set $\{0,1,2,\dots\}$ is treated as a measure space; all subsets are measurable, and the counting measure $\mu$ is used: $\mu(A)$ is the number of points in $A$ ($\infty$ if $A$ is infinite). A sequence $(a_n)$ of real numbers is just a function $a:\{0,1,2,\dots\}\to\R$ (measurable, since everything is measurable on this discrete space). It is easy to see that

• a function $a$ is integrable if and only if the series $\sum a_n$ converges absolutely, and
• in this case $\int_{\{0,1,2,\dots\}} a(n)\,\mu(\rd n) = \sum_{n=0}^\infty a_n$.

The same holds when $a_n$ are complex numbers or elements of a Banach space. In this formulation, the order of terms is evidently irrelevant, since the notion of a measure space does not stipulate any order between points. The claims about multiple and iterated series become special cases of Fubini theorem.

On the other hand, absolutely convergent series may be used when constructing Lebesgue integral.

Comments

If $\sum a_n$ and $\sum b_n$ are two convergent series, it suffices to assume the absolute convergence of just one of them to conclude that their Cauchy product also converges (see Theorem 3.50 of [Ru]). Moreover, any time that the Cauchy product of two convergent series converges, its limit must necessary be the product of the limits of the two factors (see Theorems 3.51 and 8.2 of [Ru]).

References

[Ap]	T.M. Apostol, "Mathematical analysis". Second edition. Addison-Wesley (1974) MR0344384 Zbl 0309.2600
[IlPo]	V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian) MR0687827 Zbl 0138.2730
[Ku]	L.D. Kudryavtsev, "Mathematical analysis" , 1 , Moscow (1973) (In Russian) MR0619214 Zbl 0703.26001
[Ni]	S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian) MR0466435 Zbl 0384.00004
[Ru]	W. Rudin, "Principles of mathematical analysis", Third edition, McGraw-Hill (1976) MR038502 Zbl 0346.2600