设正项级数 $\dps{\vsm{n}a_n}$ 收敛. 证明: 级数 $$\bex \vsm{n}\frac{a_n}{\sqrt{r_{n-1}}+\sqrt{r_n}} \eex$$ 仍收敛, 其中 $$\bex r_n=\sum_{k=n+1}^\infty a_k. \eex$$ (云南大学)
证明: $$\bex \vsm{n}\frac{a_n}{\sqrt{r_{n-1}}+\sqrt{r_n}} =\vsm{n}\frac{r_{n-1}-r_n}{\sqrt{r_{n-1}}+\sqrt{r_n}} =\vsm{n}\sex{\sqrt{r_{n-1}}-\sqrt{r_n}} =\sqrt{r_0}. \eex$$
