大学生数学竞赛

zhangzujin · 发表于 2023-4-26 19:48:43

2023

正项级数

1、定义: $\displaystyle \sum_{n=1}^\infty a_n, a_n\geq 0$.

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:

$$\begin{aligned} \varlimsup_{n\to\infty}\sqrt[n]{a_n}\left\{\begin{array}{llllllllllll} > 1\Rightarrow \sum_{n=1}^\infty a_n=\infty,\\\\ < 1\Rightarrow \sum_{n=1}^\infty a_n < \infty.\end{array}\right. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

(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

$$\begin{aligned} \int_1^\infty f < \infty\Leftrightarrow \sum_{n=1}^\infty f(n) < \infty. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

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

1、定义: $\displaystyle \sum_{n=1}^\infty a_n, a_n\in\mathbb{R}$.

2、敛散性:

$$\begin{aligned} \sum_{n=1}^\infty a_n\left\{\begin{array}{llllllllllll}\mbox{收敛}\left\{\begin{array}{llllllllllll}\mbox{绝对收敛}\\\\ \mbox{条件收敛}\end{array}\right.\\\\ \mbox{发散}\end{array}\right. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

3、 Abel 变换: 换种方式过把人生瘾.

$$\begin{aligned} \sum_{k=1}^n a_kb_k=A_mb_m-\sum_{n=1}^{m-1}A_n(b_{n+1}-b_n), A_n=\sum_{k=1}^n a_k. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

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

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)、 $\displaystyle \sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)}$.

2、利用子列极限.

$$\begin{aligned} 1-\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{6}-\frac{1}{8}+\cdots +\frac{1}{2n-1}-\frac{1}{4n-2}-\frac{1}{4n}+\cdots=\underline{\ \ \ \ \ \ \ \ \ \ }. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

3、求导.

$$\begin{aligned} x\in [0,\pi]\Rightarrow \sum_{n=1}^\infty \frac{\sin nx}{n}=\underline{\ \ \ \ \ \ \ \ \ \ }. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

哪个函数的正弦级数? 给我出来.