Exercise 20, § 7

Let $(u_n)$ be a sequence of finite real numbers $\geqslant 0$ such that $u_0 > 0$, and let $s_n = \sum_{k=0}^n u_k$ for each $n \geqslant 0$. Then the sequence $(u_n)$ is summable in $\mathbf{R}$ if and only if the sequence $(u_n/s_n)$ is summable in $\mathbf{R}$ [apply Theorem 4 to the sequence $(1 - u_n/s_n)]$. The same holds if the sequence $(u_n/s_n)$ is replaced by the sequence $(u_n/s_{n-1})$.