Exercise 19, § 7

Let $(1 + u_i)_{i \in I}$ be a multipliable family in $\mathbf{R}$. For each non-empty subset $H$ of $I$, let $v_H$ denote $\prod_{i \in H} u_i$. Show that the family $(v_H)_{H \in \mathfrak{P}(I)}$ is summable in $\mathbf{R}$ and that

Is the converse true?

Deduce that, if $-1 < x < 1$, then $\prod_{n=0}^\infty (1 + x^{2^n}) = 1/(1 - x)$.