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a/advanced-math/exercise/8-infinite-series/infinite-series.tex b/advanced-math/exercise/8-infinite-series/infinite-series.tex index 20b9d3c..c2eef2a 100644 --- a/advanced-math/exercise/8-infinite-series/infinite-series.tex +++ b/advanced-math/exercise/8-infinite-series/infinite-series.tex @@ -34,11 +34,132 @@ \newpage \pagestyle{plain} \setcounter{page}{1} -\section{求和函数} +\section{常数项级数} + +\subsection{正项级数} + +\subsubsection{放缩法} + +即根据收敛准则来进行判断。如果要判断原级数收敛,则辅助级数应该是对其放大,判断原级数发散,则辅助级数应该是对其缩小。 + +\subsubsection{比较判别法} + +都需要找到一个好的级数进行比较。常用的只有两个: + +$p$级数:$\sum\limits_{n=1}^\infty\dfrac{1}{n^p}\left\{\begin{array}{l} + p>1, \text{收敛} \\ + p\leqslant1, \text{发散} +\end{array}\right.$。 + +等比级数(几何级数):$\sum\limits_{n=1}^\infty\dfrac{1}aq^{n-1}\left\{\begin{array}{l} + \vert q\vert<1, \text{收敛} \\ + \vert q\vert\geqslant 1, \text{发散} +\end{array}\right.$。 + +\subsubsection{比值判别法} + +适用于含有$a^n$,$n!$,$n^n$的通项。主要是$n!$。 + +\textbf{例题:}判断$\sum\limits_{n=1}^\infty\dfrac{n!}{n^n}$。 + +解:利用比值判别法,令$a_n=\dfrac{n!}{n^n}$,$\lim\limits_{n\to\infty}\dfrac{a_{n+1}}{a_n}=\lim\limits_{n\to\infty}\dfrac{(n+1)!}{(n+1)^{n+1}}\dfrac{n^n}{n!}=\lim\limits_{n\to\infty}\dfrac{n^n}{(n+1)^n}$,注意这里幂也为变量,不是等于1而是上下同时除以$n^n$,$=\lim\limits_{n\to\infty}\dfrac{1}{(1+\frac{1}{n})^n}$,根据两个重要极限得到$=\dfrac{1}{e}<1$,所以收敛。 + +\subsubsection{根值判别法} + +适用于含有$a^n$,$n^n$的通项。 + +$\lim\limits_{n\to\infty}\sqrt[n]{n}=1$。 + +\subsubsection{积分判别法} + +\textbf{例题:}判断级数$\sum\limits_{n=2}^\infty\dfrac{1}{n\ln n}$的敛散性。 + +解:因为$\dfrac{1}{n\ln n}<\dfrac{1}{n}$,调和级数发散,所以比较判别法找不到一个较好的辅助级数。同理根据级数形式比值和根值判别法都无法使用。 + +令$f(x)=\dfrac{1}{x\ln x}$,$a_n=f(n)$,在$[2,+\infty)$上$\dfrac{1}{n\ln n}$单调减且非负。 + +级数$\sum\limits_{n=2}^\infty\dfrac{1}{n\ln n}$与$\int_2^{+\infty}\dfrac{\textrm{d}x}{x\ln x}$同敛散。 + +$=\ln\ln x\vert_2^{+\infty}=+\infty$,所以原级数发散。 + +\subsection{交错级数} + +\section{幂级数} + +\subsection{收敛域} + +\subsubsection{基本方法} + +使用比值或根值法进行求解。 + +\textbf{例题:}求幂级数$\sum\limits_{n=1}^\infty\dfrac{e^n-(-1)^n}{n^2}x^n$的收敛半径。 + +解: + +比值法: + +$\lim\limits_{n\to\infty}\left\vert\dfrac{a_{n+1}}{a_n}\right\vert=\lim\limits_{n\to\infty}\dfrac{e^{n+1}-(-1)^{n+1}}{(n+1)^2}\dfrac{n^2}{e^n-(-1)^n}=\lim\limits_{n\to\infty}\dfrac{e+(\frac{-1}{e})^n}{1-(\frac{-1}{e})^n}$。 + +又$\lim\limits_{n\to\infty}x^n=\left\{\begin{array}{ll} + 0 & \vert x\vert<1 \\ + \infty & \vert x\vert\geqslant1 +\end{array}\right.$,$\lim\limits_{n\to\infty}\left(\dfrac{-1}{e}\right)^n=0$,原式$=e$。$R=\dfrac{1}{e}$。 + +根值法: + +$\lim\limits_{n\to\infty}\sqrt[n]{\vert a_n\vert}=\lim\limits_{n\to\infty}\sqrt[n]{\dfrac{e^n-(-1)^n}{n^2}}=\lim\limits_{n\to\infty}\dfrac{e\sqrt[n]{1-(-\frac{-1}{e})^n}}{\sqrt[n]{n}\sqrt[n]{n}}$,又$\lim\limits_{n\to\infty}\sqrt[n]{n}=1$。 + +$=\lim\limits_{n\to\infty}\dfrac{e\sqrt[n]{1-0}}{1\cdot1}=e$,所以$R=\dfrac{1}{e}$。 + +\subsubsection{缺项变换} + +\textbf{例题:}求幂级数$\sum\limits_{n=1}^\infty\dfrac{n}{2^n+(-3)^n}x^{2n-1}$的收敛半径。 + +解:由于分母都是幂函数,所以使用根值法:$=\lim\limits_{n=1}^\infty\sqrt[n]{\vert a_n\vert}=\lim\limits_{n=1}^\infty\dfrac{\sqrt[n]{n}}{\sqrt[n]{3^n+(-2)^n}}\\=\lim\limits_{n=1}^\infty\dfrac{1}{\sqrt[n]{1+(-\frac{2}{3})^n}}=\dfrac{1}{3}$。 + +所以$R=3$。注意这里是错误的,因为之前求收敛域时都是$x^n$,而这里是$x^{2n-1}$,只有奇数次项,所以幂级数的一半都没有了。 + +$\sum\limits_{n=1}^\infty a_nx^{2n}\approx\sum\limits_{n=1}^\infty a_nx^{2n-1}=\sum\limits_{n=1}^\infty a_n(x^2)^n$,当前已知收敛半径为$3$,即$\vert x^2\vert<3$,即$\vert x\vert<\sqrt{3}$。 + +\subsubsection{收敛域变换} + +\textbf{例题:}已知幂级数$\sum\limits_{n=0}^\infty a_n(x+2)^n$在$x=0$处收敛,在$x=-4$处发散,求$\sum\limits_{n=0}^\infty a_n(x-3)^n$的收敛域。 + +解:根据阿贝尔定理,已知在$x=0$处收敛,且中心点在$x=-2$,则收敛区间为$(-4,0)$,在$x=-4$处发散,则$x<-4$,$x>0$处发散。 + +然后确定两端端点敛散性,$x=0$处收敛则收敛域包括$x=0$,$x=-4$处发散则收敛域不包括$x=-4$,得到收敛域$(-4,0]$。 + +对于$\sum\limits_{n=0}^\infty a_n(x-3)^n$的中心点为$x=3$,则根据相对位置收敛域为$(1,5]$。 + +\subsubsection{常数项级数变换} + +可以代入特殊点确定收敛点,将幂级数转换为常数项级数。 + +\textbf{例题:}若级数$\sum\limits_{n=0}^\infty a_n$条件收敛,求幂级数$\sum\limits_{n=0}^\infty na_n(x-1)^n$的收敛区间。 + +解:已知$\sum\limits_{n=0}^\infty a_n$条件收敛,则对于幂级数$\sum\limits_{n=0}^\infty na_n(x-1)^n$而言在$x=2$处条件收敛,即得到以中心点$x=1$的收敛区间$(0,2)$。 + +\subsection{函数展开} + +\subsubsection{因式分解} + +\textbf{例题:}将函数$f(x)=\dfrac{1}{x^2-3x-4}$展开为$x-1$的幂级数并指出收敛区间。 + +解:$\dfrac{1}{x^2-3x-4}=\dfrac{1}{5}\left(\dfrac{1}{x-4}-\dfrac{1}{x+1}\right)$。 + +$\dfrac{1}{x-4}=\dfrac{1}{(x-1)-3}=-\dfrac{1}{3}\dfrac{1}{1-\frac{x-1}{3}}=-\dfrac{1}{3}\sum\limits_{n=0}^\infty\left(\dfrac{x-1}{3}\right)^n$,$\left\vert\dfrac{x-1}{3}\right\vert<1$,$x\in(-2,4)$。 + +$\dfrac{1}{x+1}=\dfrac{1}{(x-1)+2}=\dfrac{1}{2}\dfrac{1}{1+\frac{x-1}{2}}=\dfrac{1}{2}\sum\limits_{n=0}^\infty\left(-\dfrac{x-1}{2}\right)^n$,$\left\vert-\dfrac{x-1}{2}\right\vert<1$,$x\in(-1,3)$。 + +所以其幂级数就是其加和,收敛区间为$(-2,4)\cap(-1,3)=(-1,3)$。 + +\subsection{级数求和} + +即对展开式进行逆运算,根据幂级数展开式反推原幂级数。 可以利用展开式求和函数,但是很多展开式的通项都不是公式中的,就需要对通项进行变形。 -\subsection{先导后积} +\subsubsection{先导后积} $n$在分母上,先导后积。使用变限积分:$\int_{x_0}^xS'(t)\,\textrm{d}t=S(x)-S(x_0)$,即$S(x)=S(x_0)+\int_{x_0}^xS'(t)\,\textrm{d}t$。一般选择$x_0$为展开点。 @@ -50,10 +171,9 @@ $n$在分母上,先导后积。使用变限积分:$\int_{x_0}^xS'(t)\,\textr 记$S(x)=\sum\limits_{n=1}^\infty\dfrac{x^n}{n}$,则$x^n=(x-0)^n$,取$x_0=0$。 -$\therefore S(x)=S(0)+\displaystyle{\int_0^x\left(\sum\limits_{n=1}^\infty\dfrac{t^n}{n}\right)_t'\,\textrm{d}t}=0+\int_0^x(\sum\limits_{n=1}^\infty t^{n-1})\,\textrm{d}t=\displaystyle{\int_0^x\dfrac{1}{1-t}\textrm{d}t}=-\ln(1-x)$。收敛域为$(-1,1)$。 +$\therefore S(x)=S(0)+\displaystyle{\int_0^x\left(\sum\limits_{n=1}^\infty\dfrac{t^n}{n}\right)_t'\,\textrm{d}t}=0+\int_0^x(\sum\limits_{n=1}^\infty t^{n-1})\,\textrm{d}t=\displaystyle{\int_0^x\dfrac{1}{1-t}\textrm{d}t}=-\ln(1-x)$。收敛域为$[-1,1)$。 - -\subsection{先积后导} +\subsubsection{先积后导} $n$在分子上,先积后导。$(\int S(x)\,\textrm{d}x)'=S(x)$。 @@ -61,4 +181,10 @@ $n$在分子上,先积后导。$(\int S(x)\,\textrm{d}x)'=S(x)$。 解:记$S(x)=\sum\limits_{n=1}^\infty nx^n=x\sum\limits_{n=1}^\infty x^{n-1}=x(\int\sum\limits_{n=1}^\infty nx^{n-1}\,\textrm{d}x)'=x(\sum\limits_{n=1}^\infty x^n)'=x\left(\dfrac{x}{1-x}\right)'=\dfrac{x}{(1-x)^2}$。收敛域为$[-1,1]$。 +\textbf{例题:}求级数$\sum\limits_{n=0}^\infty(n+1)(n+3)x^n$的和函数。 + +解:$(n+1)(n+3)$的形式可以推出$(n+1)(n+2)$是求两次导的结果,而这里是$(n+1)(n+3)$,所以拆开:$\sum\limits_{n=0}^\infty(n+1)(n+2)x^n+\sum\limits_{n=0}^\infty(n+1)x^n=\left(\sum\limits_{n=0}^\infty x^{n+2}\right)''+\left(\sum\limits_{n=0}^\infty x^{n+1}\right)=\left(\dfrac{x^2}{1-x}\right)''+\left(\dfrac{x}{1-x}\right)'=\dfrac{3-x}{(1-x)^3}$,$x\in(-1,1)$。 + +\section{傅里叶级数} + \end{document} diff --git a/advanced-math/knowledge/8-infinite-series/infinite-series.pdf b/advanced-math/knowledge/8-infinite-series/infinite-series.pdf index d430d119ab5b3ebfe1b2a566a313292e4b17bf13..c16c9285bb2dec2f29a791e959358539e866154e 100644 GIT binary patch delta 152988 zcmZs?Q;;THu(et0vTfToyKLLG&9`jZt}ffQZQHgr-}z%M=FH4pM&wmyL`FPo?X_oX zp>pP+;@Lr1*b_MM0fu3~+5y|(PwZESqvLWsSKkapt8L*|oDIupo7hE+SJ(v-_eGRS z4v&rf;8zG)g0gc*MpiEZSHs=9yYj2w>%VEY1o)Y)d38h%6-QZN#zBg>BpmJmc+a&R 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a/advanced-math/knowledge/8-infinite-series/infinite-series.tex +++ b/advanced-math/knowledge/8-infinite-series/infinite-series.tex @@ -62,8 +62,9 @@ $S_n=u_1+u_2+\cdots+u_n$称为级数的\textbf{部分和},$\{S_n\}$是级数 \subsubsection{性质} \begin{enumerate} - \item 线性性质:若级数$\sum\limits_{n=1}^\infty u_n$,$\sum\limits_{n=1}^\infty v_n$均收敛,且其和分别为$S$,$T$,则任给常数$a,b$,有$\sum\limits_{n=1}^\infty(au_n+bv_n)$也收敛,且其和为$aS+bT$,即$\sum\limits_{n=1}^\infty(au_n+bv_n)=a\sum\limits_{n=1}^\infty u_n+b\sum\limits_{n=1}^\infty v_n$。 + \item 线性性质:若级数$\sum\limits_{n=1}^\infty u_n$,$\sum\limits_{n=1}^\infty v_n$均收敛,且其和分别为$S$,$T$,则任给常数$a,b$,有$\sum\limits_{n=1}^\infty(au_n+bv_n)$也收敛,且其和为$aS+bT$,即$\sum\limits_{n=1}^\infty(au_n+bv_n)=a\sum\limits_{n=1}^\infty u_n+b\sum\limits_{n=1}^\infty v_n$。(收敛±发散=发散,发散±发散=不确定) \item 若级数$\sum\limits_{n=1}^\infty u_n$收敛,则其任意$m$项后余项$\sum\limits_{n=m+1}^\infty u_n$也收敛;若存在$m$项后余项$\sum\limits_{n=m+1}^\infty u_n$收敛,则$\sum\limits_{n=1}^\infty u_n$也收敛。 + \item 对收敛级数加括号仍然收敛,但是加括号收敛原级数不一定收敛。如果原级数加括号发散,则原级数发散。 \item 级数收敛必要条件:若级数$\sum\limits_{n=1}^\infty u_n$收敛,则$\lim\limits_{n\to\infty}u_n=0$。 \end{enumerate} @@ -77,6 +78,8 @@ $S_n=u_1+u_2+\cdots+u_n$称为级数的\textbf{部分和},$\{S_n\}$是级数 \textcolor{violet}{\textbf{定义:}}若通项$u_n\geqslant0$,$n=1,2,\cdots$,则$\sum\limits_{n=1}^\infty u_n$为\textbf{正项级数}。 +所以和项一定是递增的,由数列极限的单调有界准则如果和有上界则极限存在。 + \paragraph{收敛原则} \leavevmode \medskip \textcolor{aqua}{\textbf{定理:}}正项级数$\sum\limits_{n=1}^\infty u_n$收敛的充要条件是其部分和数列$\{S_n\}$有界。(某一函数在固定区间内变化率是有界的,则变化范围是有界的) @@ -87,15 +90,12 @@ $S_n=u_1+u_2+\cdots+u_n$称为级数的\textbf{部分和},$\{S_n\}$是级数 基本就是使用放缩法判断是否有界。 -\textcolor{aqua}{\textbf{定理:}}等比级数(几何级数):$\sum\limits_{n=1}^\infty\dfrac{1}aq^{n-1}\left\{\begin{array}{l} - \vert q\vert<1, \text{收敛} \\ - \vert q\vert\geqslant 1, \text{发散} -\end{array}\right.$。 - \textbf{例题:}判断级数$\sum\limits_{n=1}^\infty\dfrac{1}{\sqrt{n}}$的敛散性。 解:$S_n=1+\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+\cdots+\dfrac{1}{\sqrt{n}}>n\dfrac{1}{\sqrt{n}}=\sqrt{n}$,当$n\to\infty$时$\sqrt{n}\to\infty$,无上界所以发散。 +由于收敛原则很多时候都不能方便使用,所以出现了以下几种解决方法。 + \paragraph{比较判别法} \leavevmode \medskip \textcolor{aqua}{\textbf{定理:}}给出两个正项级数$\sum\limits_{n=1}^\infty u_n$,$\sum\limits_{n=1}^\infty v_n$,若从某项开始有$u_n\leqslant v_n$成立,则:\ding{172}若$\sum\limits_{n=1}^\infty v_n$收敛,则$\sum\limits_{n=1}^\infty u_n$也收敛;\ding{173}若$\sum\limits_{n=1}^\infty u_n$发散,则$\sum\limits_{n=1}^\infty v_n$也发散。 @@ -112,11 +112,6 @@ $S_n=\ln\dfrac{2}{1}+\ln\dfrac{3}{2}+\cdots+\ln\dfrac{n+1}{n}=\ln2-\ln1+\ln3-\ln 所以$\sum\limits_{n=1}^\infty\ln\left(1+\dfrac{1}{n}\right)$发散,则$\sum\limits_{n=1}^\infty\dfrac{1}{n}$也发散。 -\textcolor{aqua}{\textbf{定理:}}$p$级数:$\sum\limits_{n=1}^\infty\dfrac{1}{n^p}\left\{\begin{array}{l} - p>1, \text{收敛} \\ - p\leqslant1, \text{发散} -\end{array}\right.$。 - \paragraph{比较判别法极限性质} \leavevmode \medskip 是比较判别法的推论,利用极限的阶数来比较。 @@ -139,14 +134,12 @@ $\therefore\lim\limits_{n\to\infty}\dfrac{\dfrac{1}{n}-\sin\dfrac{1}{n}}{\dfrac{ \paragraph{比值判别法} \leavevmode \medskip -也称为达朗贝尔判别法。根据自己的通项的商进行比较。 +也称为达朗贝尔判别法。由等比级数推断出部分级数的敛散性只与自己的参数有关,根据自己的通项的商进行比较。 \textcolor{aqua}{\textbf{定理:}}给出一正项级数$\sum\limits_{n=1}^\infty u_n$,若$\lim\limits_{n\to\infty}\dfrac{u_{n+1}}{u_n}=\rho$,则:\ding{172}若$\rho<1$,则$\sum\limits_{n=1}^\infty u_n$收敛;\ding{173}若$\rho>1$,则$\sum\limits_{n=1}^\infty u_n$发散。 \textcolor{orange}{注意:}$\rho=1$时无法根据此判断$\sum\limits_{n=1}^\infty u_n$敛散性,如$\sum\limits_{n=1}^\infty\dfrac{1}{n}$发散,但$\sum\limits_{n=1}^\infty\dfrac{1}{n^2}$收敛。 -适用于含有$a^n$,$n!$,$n^n$的通项。 - \textbf{例题:}判断级数$\sum\limits_{n=1}^\infty\dfrac{\vert a\vert^nn!}{n^n}$的敛散性,其中$a$为非零常数。 解:记$u_n=\dfrac{\vert a\vert^nn!}{n^n}$,$\lim\limits_{n\to\infty}\dfrac{u_{n+1}}{u_n}=\vert a\vert\lim\limits_{n\to\infty}\left(\dfrac{n}{n+1}\right)^n=\vert a\vert e^{\lim\limits_{n\to\infty}n\ln\frac{n}{n+1}}=\vert a\vert e^{\lim\limits_{n\to\infty}n(\frac{n}{n+1}-1)}=\vert a\vert e^{\lim\limits_{n\to\infty}(\frac{-n}{n+1}-1)}=\vert a\vert e^{-1}=\dfrac{\vert a\vert}{e}$。 @@ -155,18 +148,24 @@ $\therefore\lim\limits_{n\to\infty}\dfrac{\dfrac{1}{n}-\sin\dfrac{1}{n}}{\dfrac{ \paragraph{根值判别法} \leavevmode \medskip -也称为柯西判别法。 +也称为柯西判别法。由比值判别法类比而来。 \textcolor{aqua}{\textbf{定理:}}给出正项级数$\sum\limits_{n=1}^\infty u_n$,若$\lim\limits_{n\to\infty}\sqrt[n]{u_n}=\rho$,则\ding{172}若$\rho<1$,则$\sum\limits_{n=1}^\infty u_n$收敛;若$\rho>1$,则$\sum\limits_{n=1}^\infty u_n$发散。 -适用于含有$a^n$,$n^n$的通项。 - 同理$\rho=1$也会失效。 \textbf{例题:}判断级数$\sum\limits_{n=1}^\infty\left(n\sin\dfrac{1}{n}\right)^{n^3}$的敛散性。 解:记$u=\left(n\sin\dfrac{1}{n}\right)^{n^3}$,则$\lim\limits_{n\to\infty}\sqrt[n]{u_n}=\lim\limits_{n\to\infty}\left(n\sin\dfrac{1}{n}\right)^{n^2}=e^{\lim\limits_{n\to\infty}n^(n\sin\frac{1}{n}-1)}=e^{\lim\limits_{n\to\infty}\dfrac{\sin\frac{1}{n}-\frac{1}{n}}{\frac{1}{n^3}}}=e^{-\frac{1}{6}}<1$,所以收敛。 +\paragraph{积分判别法} \leavevmode \medskip + +\textcolor{aqua}{\textbf{定理:}}设$f(x)$是在$[1,+\infty)$上单调递减且非负的连续函数,$a_n=f(n)$,则$\sum\limits_{n=1}^\infty a_n$与$\int_1^{+\infty}f(x)\,\textrm{d}x$同敛散。 + +\textbf{例题:}证明$p$级数$\sum\limits_{n=1}^\infty\dfrac{1}{n^p}$当$p>1$的敛散性。 + +证明:令$f(x)=\dfrac{1}{x^p}$,又$\int_1^{+\infty}\dfrac{\textrm{d}x}{x^p}$在$p>1$收敛,在$p\leqslant1$发散,所以得到原级数敛散性。 + \subsubsection{交错级数} \paragraph{概念} \leavevmode \medskip @@ -175,7 +174,9 @@ $\therefore\lim\limits_{n\to\infty}\dfrac{\dfrac{1}{n}-\sin\dfrac{1}{n}}{\dfrac{ \paragraph{莱布尼兹判别法} \leavevmode \medskip -\textcolor{violet}{\textbf{定义:}}给出一交错级数$\sum\limits_{n=1}^\infty(-1)^{n-1}u_n$,$u_n>0$,$n=1,2,\cdots$,若$\{u_n\}$\textbf{单调不增}$u_n\geqslant u_{n+1}$且$\lim\limits_{n\to\infty}=0$,则该级数收敛。 +\textcolor{violet}{\textbf{定义:}}给出一交错级数$\sum\limits_{n=1}^\infty(-1)^{n-1}u_n$,$u_n>0$,$n=1,2,\cdots$,若$\{u_n\}$\textbf{单调不增}$u_n\geqslant u_{n+1}$且$\lim\limits_{n\to\infty}=0$,则该级数收敛。反过来则不行。 + +如$\sum\limits_{n=1}^\infty\dfrac{(-1)^{n-1}}{2^{n+(-1)^n}}$收敛,但是里面的$u_n$并不递减,由根值判别法加绝对值可知$u_n$为不递减。 \textbf{例题:}判断交错调和级数$\sum\limits_{n=1}^\infty(-1)^{n-1}\dfrac{1}{n}$的敛散性。 @@ -223,6 +224,10 @@ $f'(x)=\dfrac{1-\ln(1+x)}{(1+x)^2}$,当$x\to+\infty$时,$f'(x)<0$,$\{u_n\} \textcolor{aqua}{\textbf{定理:}}若原级数绝对收敛,不论将其项如何排列,则所得的新级数也收敛,且其和不变。(绝对收敛的级数具有可交换性) +\textcolor{aqua}{\textbf{定理:}}条件收敛的级数的所有正项(或负项)构成的级数一定发散。 + +\textcolor{aqua}{\textbf{定理:}}$\sum\limits_{n=1}^\infty\vert b_n\vert$收敛,则$\sum\limits_{n=1}^\infty b_n^2$收敛。($b_n$收敛则不能得到) + \textbf{例题:}若级数$\sum\limits_{n=1}^\infty u_n$收敛,则下面级数必收敛的是()。 $A.\sum\limits_{n=1}^\infty(-1)\dfrac{u_n}{n}$\qquad$B.\sum\limits_{n=1}^\infty u_n^2$\qquad$C.\sum\limits_{n=1}^\infty(u_{2n-1}-u_{2n})$\qquad$D.\sum\limits_{n=1}^\infty(u_n+u_{n+1})$ @@ -277,7 +282,7 @@ $A.\sum\limits_{n=1}^\infty(-1)\dfrac{u_n}{n}$\qquad$B.\sum\limits_{n=1}^\infty 收敛域的求法: \begin{enumerate} - \item 若$\lim\limits_{n\to\infty}\left\vert\dfrac{a_{n+1}}{a_n}\right\vert=\rho$,则$\sum\limits_{n=0}^\infty a_nx^n$收敛半径$R$的表达式为$\left\{\begin{array}{ll} + \item 若$\lim\limits_{n\to\infty}\left\vert\dfrac{a_{n+1}}{a_n}\right\vert$或$\lim\limits_{n\to\infty}\sqrt[n]{\vert a_n\vert}=\rho$,则收敛半径$R=\left\{\begin{array}{ll} \dfrac{1}{\rho}, & \rho\neq0 \\ +\infty, & \rho=0 \\ 0, & \rho=+\infty @@ -296,6 +301,8 @@ $A.\sum\limits_{n=1}^\infty(-1)\dfrac{u_n}{n}$\qquad$B.\sum\limits_{n=1}^\infty \item 单独讨论$x=a$,$x=b$处的敛散性,得到收敛域。 \end{enumerate} +\textcolor{aqua}{\textbf{定理:}}若幂级数$\sum\limits_{n=0}^\infty a_nx^n$在点$x=x_0$处条件收敛,则点$x_0$在幂级数收敛区间的端点上。 + \textbf{例题:}求幂级数$\sum\limits_{n=1}^\infty\dfrac{x^n}{n}$的收敛域。 解:令$\vert u_n(x)\vert=\left\vert\dfrac{x^n}{n}\right\vert$。由于含有$x^n$,所以使用比值判别法。 @@ -344,40 +351,17 @@ $\sum\limits_{n=1}^\infty a_n(x+1)^n$要转换为$\sum\limits_{n=1}^\infty na_n( 而在$x=2$在收敛区间内,必然绝对收敛,所以选$A$。 -\subsection{幂级数求和函数} +\subsection{函数展开为幂级数} \subsubsection{概念} -\textcolor{violet}{\textbf{定义:}}在收敛域上,记$S(x)=\sum\limits_{n=1}^\infty u_n(x)$,并称$S(x)$为$\sum\limits_{n=1}^\infty u_n(x)$的\textbf{和函数}。 +\textcolor{violet}{\textbf{定义:}}若函数$f(x)$在$x=x_0$处存在任意阶导数,则称$f(x_0)+f'(x_0)(x-x_0)+\dfrac{f''(x_0)}{2!}(x-x_0)^2+\cdots+\dfrac{f^{(n)}(x_0)}{n!}(x-x_0)^n+\cdots$为函数$f(x)$在$x_0$处的\textbf{泰勒级数},则$f(x)=\sum\limits_{n=0}^\infty\dfrac{f^{(n)}(x_0)}{n!}(x-x_0)^n$。 -\subsubsection{运算法则} +当$x_0=0$时,称$f(0)+f'(0)x+\dfrac{f''(0)}{2!}x^2+\cdots+\dfrac{f^{(n)}(0)}{n!}x^n+\cdots$为函数$f(x)$的\textbf{麦克劳林级数},若收敛,则$f(x)=\sum\limits_{n=0}^\infty\dfrac{f^{(n)}(0)}{n!}x^n$。 -若幂级数$\sum\limits_{n=0}^\infty a_nx^n$与$\sum\limits_{n=0}^\infty b_nx^n$的收敛半径分别为$R_a$和$R_b$($R_a\neq R_b$),则: +都是函数展开成幂级数。 -\begin{itemize} - \item $k\sum\limits_{n=0}^\infty a_nx^n=\sum\limits_{n=0}^\infty ka_nx^n$,$\vert x\vert