Let $\{a_n\}$ be a sequence. We say $\{a_n\}$ is summable if the sequence $\{s_n\}$ defined by $s_1=a_1\\s_{n+1}=s_n+a_{n+1}$
converges to $\ell\in\Bbb R$, and write $\sum\limits_{n = 1}^\infty {{a_n}} = \ell $
$(1)$ Suppose $\{a_n\}$ is of non-negative terms, and that it is not summable, that is $\lim {s_n}$ fails to exist. Show that $\left\{ {\frac{{{a_n}}}{{1 + {a_n}}}} \right\}$ is not summable either.
Now, since $0 \leqslant \frac{{{a_n}}}{{1 + {a_n}}} \leqslant {a_n}$ for each $n$, $\{a_n\}$ being summable implies $\left\{ {\dfrac{{{a_n}}}{{1 + {a_n}}}} \right\}$ also is (monotone convergence). I need to show the converse of this, but I can see no way to prove it. Since $a_n\geq0$, the partial sums ${s_N} = \sum\limits_{n = 1}^N {{a_n}} $can be made as large as we want.