Proposition 8 (Restricted associativity of series)

Let $(k_n)$ be a strictly increasing sequence of integers $\geqslant 0$. If the series whose general term is $x_n$ converges, and if we put $u_n = \sum_{p=k_{n-1}}^{k_n-1} x_p$, then the series whose general term is $u_n$ is convergent, and we have $\sum_{n=0}^{\infty} u_n = \sum_{n=0}^{\infty} x_n$.

For the sequence of partial sums of the series $(u_n)$ is a subsequence $(s_{k_n-1})$ of the sequence $(s_n)$ of partial sums of the series $(x_n)$.