无穷级数
2022-05-16 16:05:08 0 举报AI智能生成
浙江省专升本数学-无穷级数
作者其他创作
大纲/内容
常数项级数的概念和性质
常数项级数的概念
定义1
如果给定一个数列<span class="equation-text" contenteditable="false" data-index="0" data-equation="u_1,u_2,u_3,\cdots,u_n,\cdots,则由这数列构成的表达式u_1+u_2+u_3+\cdots+u_n+\cdots叫做(常数项)无穷级数，记为\sum _{n=1}^{\propto} u_n,其中第n项u_n叫做级数的一般项。"><span></span><span></span></span>
定义2
作级数前<span class="equation-text" contenteditable="false" data-index="0" data-equation="n"><span></span><span></span></span>项的和<span class="equation-text" contenteditable="false" data-index="1" data-equation="s_n"><span></span><span></span></span>,<span class="equation-text" contenteditable="false" data-index="2" data-equation="s_n"><span></span><span></span></span>称为级数<span class="equation-text" contenteditable="false" data-index="3" data-equation="\sum _{n=1}^{\propto}u_n的部分和"><span></span><span></span></span>
定义3
<span class="equation-text" contenteditable="false" data-index="0" data-equation="如果级数的部分和数列\{s_n\}有极限s,即\lim _{n\rightarrow \propto} s_n = s,称为无穷级数收敛，这时极限s叫做这级数的和，并写成s=u_1+u_2+u_3+\cdots+u_n+\cdots;如果\{s_n\}没有极限，则称为无穷级数发散"><span></span><span></span></span>
两个重要级数
几何级数(等比级数)
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sum _{n=1}^{\propto}aq^{n-1}=a+aq+aq^2+\cdots+aq^{n-1}+\cdots(a\neq 0)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="|q|<1，级数收敛，其和为\frac{a}{1-q}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="|q|\geq 1，级数发散"><span></span><span></span></span>
讨论
<span class="equation-text" contenteditable="false" data-index="0" data-equation="q=1时，S_n=na,\lim_{n\rightarrow \propto} S_n = \lim_{n\rightarrow \propto} na = \propto,级数发散"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="q=-1时，S_n=a-a+a-a+\cdots,\lim_{n\rightarrow \propto} S_n不存在，级数发散"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="q\neq 1时，S_n=\frac{a(1-q^n)}{1-q}=\begin{cases}|q|<1,\lim_{n\rightarrow \propto} S_n=\frac{a}{1-q} ，级数收敛\\|q|>1,\lim_{n\rightarrow \propto} S_n=\propto，级数发散\end{cases}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="P级数"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sum _{n=1}^{\propto} \frac{1}{n^p} =1+\frac{1}{2^p} +\frac{1}{3^p}+\cdots +\frac{1}{n^p}+\cdots(p>0)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="p>1,级数收敛"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="p<1，级数发散"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="p=1时，调和级数\sum _{n=1}^{\propto} \frac{1}{n}发散"><span></span><span></span></span>
广义P级数
收敛级数的基本性质
性质1
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若级数\sum _{n=1}^{\propto}u_n 收敛且和为s，则级数\sum _{n=1}^{\propto}ku_n也收敛，且和为ks"><span></span><span></span></span>
性质2
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若级数\sum _{n=1}^{\propto}u_n、\sum _{n=1}^{\propto}v_n分别收敛于s、\delta，则级数\sum _{n=1}^{\propto}(u_n\pm v_n)也收敛，且其和为s\pm \delta"><span></span><span></span></span>
性质3
<span class="equation-text" contenteditable="false" data-index="0" data-equation="在级数中改变(去掉、加上或改变)有限项，不会改变级数的敛散性"><span></span><span></span></span>
性质4
收敛级数加括号后所成的级数仍然收敛于原级数的和，但加括号后所成的级数收敛，去掉括号后原级数未必收敛
性质5
(级数收敛的必要条件)如果级数<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sum _{n=1}^{\propto}u_n收敛，则它的一般项u_n趋于零，即\lim _{n\rightarrow \propto} u_n =0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="非充分的例子：调和级数 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n} +\cdots的一般项u_n=\frac{1}{n} \rightarrow 0 (0 \rightarrow \propto) ,但是\sum _{n=1}^{\propto}\frac{1}{n}是发散的"><span></span><span></span></span>
注
收敛+收敛=收敛，收敛+发散=发散
常数项级数的审敛法
正项级数及其审敛法
定义1
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若u_n \geq 0 ,则\sum _{n=1}^{\propto}u_n称为正项级数"><span></span><span></span></span>
定理1
正项级数收敛的充要条件是部分和数列<span class="equation-text" contenteditable="false" data-index="0" data-equation="\{S_n\}有界"><span></span><span></span></span>
（定理2、3、4）正项级数审敛法(<span class="equation-text" contenteditable="false" data-index="0" data-equation="对一切自然数n，都有u_n\geq 0 ,称级数\sum _{n=1}^{\propto}u_n为正项级数"><span></span><span></span></span>)
比较审敛法（大敛则小敛，小散则大散）
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设\sum _{n=1}^{\propto}u_n和\sum _{n=1}^{\propto}v_n都是正项级数，且u_n\leq v_n (n=1,2,\cdots)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若\sum _{n=1}^{\propto}v_n收敛，则级数\sum _{n=1}^{\propto}u_n收敛"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若\sum _{n=1}^{\propto}u_n发散，则级数\sum _{n=1}^{\propto}v_n发散"><span></span><span></span></span>
注
<span class="equation-text" contenteditable="false" data-index="0" data-equation="当x>0时，sinx<x"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="1" data-equation="ln(1+x)<x"><span></span><span></span></span>
比较审敛法的极限形式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若\lim_{n\rightarrow \propto } \frac{u_n}{v_n} =l(0\leq l<+\propto) "><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}l>0,同敛散 \\l=0,母收子收 \\l=+\propto,母散子散\end{cases}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="本质就是将u_n放大到v_n,就是放缩法"><span></span><span></span></span>
比值审敛法(达朗贝尔判别法，通常用在<span class="equation-text" contenteditable="false" data-index="0" data-equation="n!、a^n、n^n"><span></span><span></span></span>)
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若\lim_{n\rightarrow \propto } \frac{u_{n+1}}{u_n} =\rho"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\rho<1,级数收敛"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\rho>1(包括\lim_{n\rightarrow \propto} \frac{u_{n+1}}{u_n}=\propto),级数发散"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\rho = 1,方法失效"><span></span><span></span></span>
根值审敛法
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若\lim_{n\rightarrow \propto} \sqrt[n]{u_n} =\rho"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\rho<1,级数收敛"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\rho>1(包括\lim_{n\rightarrow \propto} \frac{u_{n+1}}{u_n}=\propto),级数发散"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\rho = 1,方法失效"><span></span><span></span></span>
定理5
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设级数\sum _{n=1}^{\propto} u_n 为正项级数"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="如果\lim_{n\rightarrow \propto} n\cdot u_n=l>0(或\lim_{n\rightarrow \propto} n\cdot u_n=+\propto),则级数\sum_{n=1}^{\propto} u_n发散"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="如果p>1,而\lim_{n\rightarrow \propto} n^p\cdot u_n = l (0\leq l < +\propto),则级数\sum_{n=1}^{\propto} u_n收敛"><span></span><span></span></span>
交错级数及其审敛法
定义1
所谓交错级数就是这样的级数，它的各项是征服交错的，从而可以写成下面的形式：<span class="equation-text" contenteditable="false" data-index="0" data-equation="u_1-u_2+u_3-u_4+\cdots或-u_1+u_2-u_3+u_4-\cdots其中u_1,u_2\cdots都是正数"><span></span><span></span></span>
定理7（交错级数的莱布尼兹审敛法）
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设\sum _{n=1}^{\propto} (-1)^{n-1} u_n ,u_n>0为交错级数"><span></span><span></span></span>
满足收敛的两个条件
<span class="equation-text" contenteditable="false" data-index="0" data-equation="对一切n有u_{n+1}\leq u_n(单调递减)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{n\rightarrow \propto} u_n =0"><span></span><span></span></span>
绝对收敛和条件收敛（通常和比较审敛法一起用）
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若\sum _{n=1}^{\propto} |u_n| 收敛，则级数\sum _{n=1}^{\propto} u_n绝对收敛， "><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若\sum _{n=1}^{\propto} u_n收敛，而\sum _{n=1}^{\propto} |u_n| 发散，则级数\sum _{n=1}^{\propto} u_n条件收敛 "><span></span><span></span></span>
幂级数
函数项级数的概念
如果一个定义在区间I上的函数数列<span class="equation-text" contenteditable="false" data-index="0" data-equation="u_1(x),u_2(x),u_3(x),\cdots,u_n(x),\cdots则由着函数数列构成的表达式u_1(x)+u_2(x)+u_3(x)+\cdots+u_n(x)+\cdots称为定义域在区间I上的(函数项)无穷级数，简称(函数项)级数"><span></span><span></span></span>
幂级数及其收敛性
幂级数的概念
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设a_n(n=0,1,2,\cdots)为常数，则形如\sum _{n=0}^{\propto} a_n(x-x_0)^n的级数称为x-x_0的幂级数，常数a_n(n=0,1,2,3,\cdots)称为幂级数的系数，当x_0=0时\sum _{n=0}^{\propto} a_nx^{n}称为x的幂级数"><span></span><span></span></span>
阿贝尔定理
<span class="equation-text" contenteditable="false" data-index="0" data-equation="对于任意一个幂级数\sum _{n=0}^{\propto} a_nx^{n} ,都存一个R，0\leq R\leq+\propto，对于一切|x|<R都有级数\sum _{n=0}^{\propto} a_nx^n绝对收敛"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="而当|x|>R时级数发散。称为R为该幂级数的收敛半径，（-R,R）为收敛区间，当幂级数只在x=0一点收敛时，R=0，当对于一切x幂级数都收敛时R=+\propto"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(-\propto,-R)(-R,R)(R,+\propto)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="当x=R与x=-R时候，幂级数可能收敛也可能发散"><span></span><span></span></span>
收敛半径、区间的求法
对于幂级数<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sum _{n=1}^{\propto} u_n(x)，且\rho=\lim_{n\rightarrow \propto } |\frac{u_{n+1}(x)}{u_n(x)}|=\lambda(x),则\lambda(x)<1的解即为幂级数的收敛区间"><span></span><span></span></span>,<span class="equation-text" contenteditable="false" data-index="1" data-equation="R=\frac{1}{\rho}，\lambda(x)=0,R=+\propto,\lambda(x)=+\propto,R=0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="收敛半径=\frac{收敛区间长度}{2}"><span></span><span></span></span>
收敛域=收敛区间+收敛端点
<span class="equation-text" contenteditable="false" data-index="0" data-equation="幂级数\sum _{n=0}^{\propto} a_n\cdot x^{n} 的收敛半径R=\lim_{n\rightarrow \propto} |\frac{a_n}{a_{n+1}}|"><span></span><span></span></span>
注
R=0时,仅在x=0处收敛；<span class="equation-text" contenteditable="false" data-index="0" data-equation="R=+\propto,"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="(-\propto,+\propto)处收敛"><span></span><span></span></span>
幂级数性质
运算性质
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设幂级数\sum _{n=0}^{\propto} a_nx^n收敛半径为R_1，和函数为S_1(x),而幂级数\sum _{n=0}^{\propto} b_nx^n的收敛半径为R_2，和函数为S_2(x)，令R=min(R1,R2)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sum _{n=0}^{\propto} a_nx^n\pm\sum _{n=0}^{\propto} b_nx^n=\sum _{n=0}^{\propto} (a_n\pm b_n)x^n=S_1(x)\pm S_2(x),x\in (-R,R)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\sum _{n=0}^{\propto} a_nx^n)(\sum _{n=0}^{\propto} b_nx^n)=\sum _{n=0}^{\propto} (a_0b_n+a_1b_{n-1+\cdots+a_nb_0})x^n=S_1(x) S_2(x),x\in (-R,R)"><span></span><span></span></span>
和函数性质
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设\sum _{n=0}^{\propto} a_nx^n收敛半径为R，和函数S(x)，则"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="S(x)在(-R,R)内连续"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="S(x)在(-R,R)内可导，且可逐项求导，即S^{'}(x)=\sum _{n=1}^{\propto} na_nx^{n-1}，收敛半径不变"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="S(x)在(-R,R)上可积，且逐项可积，即\int _0^xS(x)dx=\sum _{n=0}^{\propto} \int _0^x a_nx^n = \sum _{n=0}^{\propto} \frac{a_nx^{n+1}}{n+1}，，收敛半径不变"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="消倍数，先积分后求导"><span></span><span></span></span>
S(0)可加可不加<br>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="消分数，先求导后积分"><span></span><span></span></span>
记住要加上S(0),计算S(0)时候，先展开在代入值
注
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sum _{n=0}^{\propto} x^n=1+x+x^2+\cdots+x^n+\cdots=\lim_{n\rightarrow \propto}\frac{1\cdot(1-x^n)}{1-x}=\begin{cases}\frac{1}{1-x} ,|x|<1\\发散，|x|\geq1\end{cases}"><span></span><span></span></span>
函数展开成幂级数
泰勒级数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设f(x)在x=x_0处任意阶可导，则幂级数\sum _{n=0}^{\propto} \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n=f(x_0)+f^{'}(x_0)(x-x_0)+\cdots+\frac{f^{n}(x_0)}{n!}(x-x_0)^n+\cdots称为f(x)在x=x_0处的泰勒级数"><span></span><span></span></span>
麦克劳林级数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="当x_0=0时，级数\sum _{n=0}^{\propto} \frac{f^{(n)}(0)}{n!}x^n=f(0)+f^{'}(0)x+\cdots+\frac{f^{n}(0)}{n!}x^n+\cdots称为f(x)的麦克劳林级数"><span></span><span></span></span>
泰勒级数的收敛定理
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设f(x)在x=x_0处任意阶可导，则泰勒级数\sum _{n=0}^{\propto} \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n收敛于f(x)的充要条件是\lim_{n\rightarrow \propto}R(x)=0,其中"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="R_n=\frac{f^{(n+1)}[x_0+\theta(x-x_0)]}{(n+1)!}(x-x_0)^{n+1},0<\theta<1"><span></span><span></span></span>
常用的麦克劳林级数
注
展开的方法就是将所给式子在保证凑成展开的样子之后，朝着常用的麦克劳林式子凑
当所给的<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)和展成的之间差了次方可以尝试积分或者求导回到展开的样子再积分或求导回去"><span></span><span></span></span>
0 条评论
下一页
为你推荐
查看更多
