\[
\displaylines{\text{Suppose }a_{n}\gt0,s_{n}=a_1+\cdots+a_{n},\text{ and }\sum a_{n}\text{ diverges.}\\ 1)\text{Prove that }\sum\frac{a_{n}}{1+a_{n}}\text{ diverges.}\\ 2)\text{Prove that }\frac{a_{N+1}}{s_{N+1}}+\cdots+\frac{a_{N+k}}{s_{N+k}}\ge1-\frac{s_{N}}{s_{N+k}}\\ \text{and deduce that }\sum\frac{a_{n}}{s_{n}}\text{ diverges.}\\ 3)\text{Prove that }\frac{a_{n}}{s_{n}^2}\le\frac{1}{s_{n-1}}-\frac{1}{s_{n}}\\ \text{and deduce that }\sum\frac{a_{n}}{s_{n}^2}\text{ converges.}}
\]