Proposition 4

A series of finite real numbers is commutatively convergent if and only if it is absolutely convergent.

This follows from Chapter III, § 5, no. 7, Proposition 9 and from Theorem 3 of no. 2.

In other words, if $(u_n)$ is a sequence of finite real numbers, it comes to the same thing to say that the series whose general term is $u_n$ is commutatively convergent, or that it is absolutely convergent, or that the sequence $(u_n)$ is summable in $\mathbf{R}$. All the properties of summable families proved in Chapter III, § 5 therefore apply to absolutely convergent series. In particular, if the series whose general term is $u_n$ is absolutely convergent, then the sum $\sum_{n \in H} u_n$ exists for all subsets $H$ of $\mathbf{N}$; and, if $(H_p)$ is a partition of $\mathbf{N}$, we have

(associativity of absolutely convergent series).

As we have already remarked in Chapter III, § 5, a series of real numbers can be convergent without being commutatively convergent, that is, without being absolutely convergent.