📝 题目
4. 1.25 (1) 设 $0 < q < 1$ ,求证: $\exists r \in \left( {q,1}\right)$ ,使 $n$ 充分大时,有
$$ 1 + \frac{q}{n} < {\left( 1 + \frac{1}{n}\right) }^{r}\;\left( {n > N}\right) ; $$
(2)设 ${a}_{n} > 0$ ,求证: $\mathop{\lim }\limits_{{n \rightarrow \infty }}n\left( {\frac{1 + {a}_{n + 1}}{{a}_{n}} - 1}\right) \geq 1$ .
💡 答案与解析
4. 1.25 (2) 用反证法证得级数 $\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{1}{{\left( n + 1\right) }^{r}}\left( {0 < r < 1}\right)$ 收敛,从而得矛盾.