大学生数学竞赛
# 2023
## 正项级数
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1、 定义:$\displaystyle \sum_{n=1}^\infty a_n, a_n\geq 0$.
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2、 审敛法. 本节所涉级数均为正项级数.
(1)、 与别人比较:难易选取榜样. 比较判别法:
$$\begin{aligned} \lim_{n\to\infty}\frac{a_n}{b_n}=\ell\in [0,\infty),\sum_{n=1}^\infty b_n<\infty\Rightarrow \sum_{n=1}^\infty a_n<\infty,\\\\ \lim_{n\to\infty}\frac{a_n}{b_n}=\ell\in (0,\infty), \sum_{n=1}^\infty b_n=+\infty\Rightarrow \sum_{n=1}^\infty a_n=+\infty. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$
(2)、 与自己的过往比较:自省好样的. 比值判别法 (d'Alembert):
$$\begin{aligned} \varlimsup_{n\to\infty}\frac{a_{n+1}}{a_n}\left\{\begin{array}{llllllllllll}<1\Rightarrow \sum_{n=1}^\infty a_n<\infty,\\\\> 1\Rightarrow \sum_{n=1}^\infty a_n=+\infty.\end{array}\right. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$
(3)、 活在当下就好:别瞎操心了. Cauchy 判别法:
$$\begin{aligned} \varlimsup_{n\to\infty}\sqrt{a_n}\left\{\begin{array}{llllllllllll} > 1\Rightarrow \sum_{n=1}^\infty a_n=\infty,\\\\ <1\Rightarrow \sum_{n=1}^\infty a_n<\infty.\end{array}\right. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$
(4)、 榜样的力量. Rabbe 判别法:
$$\begin{aligned} \lim_{n\to\infty}n\left(\frac{a_n}{a_{n+1}}-1\right)\left\{\begin{array}{llllllllllll} > 1\Rightarrow \sum_{n=1}^\infty a_n<\infty,\\\\ <1\Rightarrow \sum_{n=1}^\infty a_n=\infty.\end{array}\right. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$
(5)、 单调人生其实也有趣:知微见著. Cauchy 判别法:设 $\displaystyle a_n\searrow, \geq 0$, 则
$$\begin{aligned} \sum_{n=1}^\infty a_n<\infty\Leftrightarrow \sum_{n=0}^\infty 2^n a_{2^n}<\infty. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$
(5-1)、 考查 $\displaystyle \sum_{n=1}^\infty \frac{1}{n^k}$.
(5-2)、 考查 $\displaystyle \sum_{n=1}^\infty \frac{1}{n\ln^pn}$.
(6)、 单调人生其实也有趣:圆滑与耿直集一身. 设 $\displaystyle f\searrow, \geq 0$, 则
$$\begin{aligned} \int_1^\infty f<\infty\Leftrightarrow \sum_{n=1}^\infty f(n)<\infty. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$
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3、 万古垂青与遗臭万年都是极品. 一般人就过日子呗. 人生代代无穷已, 江月年年只相似.
(1)、 若 $\displaystyle \sum_{n=1}^\infty a_n<\infty$, 则存在 $\displaystyle b_n:\sum_{n=1}^\infty b_n<\infty, \lim_{n\to\infty}\frac{a_n}{b_n}=0$.
(2)、 若 $\displaystyle \sum_{n=1}^\infty a_n=\infty$, 则存在 $\displaystyle b_n:\sum_{n=1}^\infty b_n=\infty, \lim_{n\to\infty}\frac{b_n}{a_n}=0$.
## 一般项级数
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1、 定义:$\displaystyle \sum_{n=1}^\infty a_n, a_n\in\mathbb{R}$.
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2、 敛散性:
$$\begin{aligned} \sum_{n=1}^\infty a_n\left\{\begin{array}{llllllllllll}\mbox{收敛}\left\{\begin{array}{llllllllllll}\mbox{绝对收敛}\\\\ \mbox{条件收敛}\end{array}\right.\\\\ \mbox{发散}\end{array}\right. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$
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3、 Abel 变换:换种方式过把人生瘾.
$$\begin{aligned} \sum_{k=1}^n a_kb_k=A_mb_m-\sum_{n=1}^{m-1}A_n(b_{n+1}-b_n), A_n=\sum_{k=1}^n a_k. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$
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4、 Dirichlet 有两个小宝宝:Abel 与 Leibniz.
(1)、 Dirichlet:$\displaystyle S_n=\sum_{k=1}^n a_k$ 有界, $\displaystyle b_n\searrow 0$, 则 $\displaystyle \sum_{n=1}^\infty a_nb_n$ 收敛.
(2)、 Abel:$\displaystyle \sum_{n=1}^\infty a_n$ 收敛, $\displaystyle b_n$ 单调有界, 则 $\displaystyle \sum_{n=1}^\infty a_nb_n$ 收敛.
(3)、 Leibniz:$\displaystyle a_n\searrow 0\Rightarrow \sum_{n=1}^\infty (-1)^na_n$ 收敛.
## 求和问题
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1、 裂项相消.
(1)、 $\displaystyle 0<x<1\Rightarrow \sum_{n=0}^\infty \frac{x^{2^n}}{1-x^{2^{n+1}}}=\underline{\\\\\\\\\\}$.
(2)、 $\displaystyle \sum_{k=1}^\infty \arctan \frac{1}{2k^2}$.
(3)、 $\displaystyle \sum_{k=2}^\infty \arctan \frac{2}{4k^2-4k+1}$.
(4)、 $\displaystyle \sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)}$.
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2、 利用子列极限.
$$\begin{aligned} 1-\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{6}-\frac{1}{8}+\cdots +\frac{1}{2n-1}-\frac{1}{4n-2}-\frac{1}{4n}+\cdots=\underline{\\\\\\\\\\}. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$
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3、 求导.
$$\begin{aligned} x\in \Rightarrow \sum_{n=1}^\infty \frac{\sin nx}{n}=\underline{\\\\\\\\\\}. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$
哪个函数的正弦级数? 给我出来.
# 2022
## 01数列极限
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1、 第01-09题
(1)、 https://www.bilibili.com/video/BV1qN411N7P7
(2)、 https://www.bilibili.com/video/BV1YN411N7Ee
(3)、 https://www.bilibili.com/video/BV1Sj411w7AL
(4)、 https://www.bilibili.com/video/BV1ms4y1U7eT
(5)、 https://www.bilibili.com/video/BV1ds4y177wr
(6)、 https://www.bilibili.com/video/BV1ac41157ua
(7)、 https://www.bilibili.com/video/BV1jg4y1G74J
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2、 第10-20题
(1)、 https://www.bilibili.com/video/BV1bs4y1U7EF
(2)、 https://www.bilibili.com/video/BV15X4y1d7HQ
(3)、 https://www.bilibili.com/video/BV1nV4y1S749
(4)、 https://www.bilibili.com/video/BV1Qs4y1772X
(5)、 https://www.bilibili.com/video/BV1ya4y1M786
(6)、 https://www.bilibili.com/video/BV1tm4y1z7qV
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3、 第21-27题
(1)、 https://www.bilibili.com/video/BV1Eg4y1G7qb
(2)、 https://www.bilibili.com/video/BV1ha4y1K71k
(3)、 https://www.bilibili.com/video/BV12T411s7pu
(4)、 https://www.bilibili.com/video/BV1kT411s76f
## 02函数极限
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4、 第01-07题
(1)、 https://www.bilibili.com/video/BV1hV4y1S7tJ
(2)、 https://www.bilibili.com/video/BV1HN411A72P
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5、 第08-14题
(1)、 https://www.bilibili.com/video/BV1dh411u7Xe
## 03微分
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6、 第01-12题
(1)、 https://www.bilibili.com/video/BV1RT411s7vy
(2)、 https://www.bilibili.com/video/BV17o4y1p78S
(3)、 https://www.bilibili.com/video/BV1HL411U78W
(4)、 https://www.bilibili.com/video/BV1Mk4y1v7s7
(5)、 https://www.bilibili.com/video/BV1pM411T7dk
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7、 第13-21题
(1)、 https://www.bilibili.com/video/BV1xM411T7QD
(2)、 https://www.bilibili.com/video/BV1iN411A75X
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