[裴礼文数学分析中的典型问题与方法习题参考解答]5.1.9

证明: 若有 $\al>0$, 使当 $n\geq n_0$ 时, $\dps{\frac{\ln \frac{1}{a_n}}{\ln n}\geq 1+\al\ (a_n>0)}$, 则级数 $\dps{\vsm{n}a_n\ (a_n>0)}$ 收敛; 若 $n\geq n_0$ 时, $\dps{\frac{\ln \frac{1}{a_n}}{\ln n}\leq 1}$, 则这级数发散 (对数判别法).

证明:

(1). $$\bex \frac{\ln \frac{1}{a_n}}{\ln n}\geq 1+\al\ra \ln \frac{1}{a_n}\geq \ln n^{1+\al} \ra a_n\leq \frac{1}{n^{1+\al}}\ (\al>0). \eex$$

(2). $$\bex \frac{\ln \frac{1}{a_n}}{\ln n}\leq 1\ra a_n\geq \frac{1}{n}. \eex$$