1. 证明: 数列 $\{a_n\}$ 收敛当且仅当其奇子列和偶子列收敛于同一极限.
Posted by haifeng on 2023-12-17 23:38:13 last update 2023-12-17 23:41:15 | Answers (0) | 收藏
证明: 数列 $\{a_n\}$ 收敛当且仅当其奇子列和偶子列收敛于同一极限.
梅加强《数学分析》P.36, 习题17.
Posted by haifeng on 2023-12-17 23:38:13 last update 2023-12-17 23:41:15 | Answers (0) | 收藏
证明: 数列 $\{a_n\}$ 收敛当且仅当其奇子列和偶子列收敛于同一极限.
梅加强《数学分析》P.36, 习题17.
Posted by haifeng on 2023-11-07 19:24:20 last update 2023-11-07 19:24:42 | Answers (0) | 收藏
Posted by haifeng on 2023-11-03 08:30:21 last update 2023-11-03 08:30:21 | Answers (1) | 收藏
求下列极限.
\[
\lim_{x\rightarrow 0}\Bigl(\frac{1}{x}-\frac{1}{e^x-1}\Bigr)
\]
Posted by haifeng on 2023-10-20 21:06:44 last update 2023-10-20 21:07:12 | Answers (2) | 收藏
设 $x_1=\sqrt{6}$, $x_{n+1}=\sqrt{6+x_n}$, $n=1,2,3,\ldots$. 证明数列 $\{x_n\}$ 收敛, 并求其极限.
Posted by haifeng on 2023-10-14 09:15:35 last update 2023-10-14 09:15:35 | Answers (1) | 收藏
设 $a,b$ 是常数, 满足 $\lim\limits_{x\rightarrow 1}\frac{\sqrt{x+a}+b}{x^2-1}=1$, 求 $a$ 和 $b$ 的值.
Posted by haifeng on 2023-10-12 08:07:58 last update 2023-10-12 08:07:58 | Answers (1) | 收藏
求极限 $\lim\limits_{x\rightarrow\infty}\Bigl(\dfrac{2x-4}{2x+1}\Bigr)^{2x}$.
Posted by haifeng on 2023-10-11 23:28:21 last update 2023-10-11 23:28:21 | Answers (1) | 收藏
求极限 $\lim\limits_{x\rightarrow 0}\dfrac{\sec x-\cos x}{x^2}$.
Posted by haifeng on 2023-10-11 19:23:42 last update 2023-10-11 19:24:05 | Answers (1) | 收藏
利用极限存在准则证明下列极限.
(1) $\lim\limits_{n\rightarrow\infty}\dfrac{n!}{n^n}=0$.
(2) $\lim\limits_{n\rightarrow\infty}\Bigl(1+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\cdots+\dfrac{1}{n^2}\Bigr)$ 存在.
Posted by haifeng on 2023-10-11 16:13:32 last update 2023-10-11 16:13:43 | Answers (1) | 收藏
利用极限的定义证明
(1) $\lim\limits_{x\rightarrow\infty}\dfrac{x+1}{2x}=\dfrac{1}{2}$
(2) $\lim\limits_{x\rightarrow\infty}\dfrac{x^2-1}{x-1}=2$