Exercise 1, § 7

Let $(x_i)$ be a family of real numbers, all of which belong to the interval $A' = ]-\infty, +\infty]$ (resp. $A'' = [-\infty, +\infty[$). The family $(x_i)$ is summable in $\overline{\mathbf{R}}$ if and only if one of the following conditions is satisfied: a) At least one of the numbers $x_i$ is equal to $+\infty$ (resp. $-\infty$). b) At least one of the two sums $\sum_i x_i^+$, $\sum_i x_i^-$ is finite.

In the first case we have $\sum_i x_i = +\infty$ (resp. $-\infty$); in the second case, $\sum_i x_i = \sum_i x_i^+ - \sum_i x_i^-$.