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b/advanced-math/knowledge/8-infinite-series/infinite-series.tex index 3c4b1e9..3b6c717 100644 --- a/advanced-math/knowledge/8-infinite-series/infinite-series.tex +++ b/advanced-math/knowledge/8-infinite-series/infinite-series.tex @@ -210,6 +210,8 @@ $f'(x)=\dfrac{1-\ln(1+x)}{(1+x)^2}$,当$x\to+\infty$时,$f'(x)<0$,$\{u_n\} 给任意项级数每一项加上绝对值$\sum\limits_{n=1}^\infty(-1)^{n-1}\vert u_n\vert$,就得到了正项级数,称为原级数的\textbf{绝对值级数}。 +添加绝对值会提高发散性,因为不改变逼近0的速度,而让各项之间不能抵消为0。 + \paragraph{绝对收敛} \leavevmode \medskip \textcolor{violet}{\textbf{定义:}}设$\sum\limits_{n=1}^\infty(-1)^{n-1}u_n$为任意项级数,若$\sum\limits_{n=1}^\infty(-1)^{n-1}\vert u_n\vert$收敛,则称$\sum\limits_{n=1}^\infty$\\$(-1)^{n-1}u_n$\textbf{绝对收敛}。 @@ -226,7 +228,7 @@ $f'(x)=\dfrac{1-\ln(1+x)}{(1+x)^2}$,当$x\to+\infty$时,$f'(x)<0$,$\{u_n\} \textcolor{aqua}{\textbf{定理:}}条件收敛的级数的所有正项(或负项)构成的级数一定发散。 -\textcolor{aqua}{\textbf{定理:}}$\sum\limits_{n=1}^\infty\vert b_n\vert$收敛,则$\sum\limits_{n=1}^\infty b_n^2$收敛。($b_n$收敛则不能得到) +\textcolor{aqua}{\textbf{定理:}}$\sum\limits_{n=1}^\infty\vert a_n\vert$收敛,则$\sum\limits_{n=1}^\infty a_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()。 @@ -337,6 +339,8 @@ $\therefore\lim\limits_{n\to\infty}\dfrac{\vert u_{n+1}(x)\vert}{\vert u_n(x)\ve \end{enumerate} \end{enumerate} +收敛域的扩大和缩小在于其端点是否通过求导或积分变得可取了。 + \textbf{例题:}设$\sum\limits_{n=1}^\infty a_n(x+1)^n$在点$x=1$处条件收敛,则幂级数$\sum\limits_{n=1}^\infty na_n(x-1)^n$在点$x=2$处()。 $A.$绝对收敛\qquad$B.$条件收敛\qquad$C.$发散\qquad$D.$敛散性不确定 @@ -403,6 +407,8 @@ $x$的取值指其幂指数的收敛域。第七个幂函数问题较复杂, \subsection{幂级数求和函数} +即函数展开的逆操作。 + \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}。 @@ -429,23 +435,125 @@ $x$的取值指其幂指数的收敛域。第七个幂函数问题较复杂, \subsubsection{性质} -收敛域的扩大和缩小在于其端点是否通过求导或积分变得可取了。 - \begin{itemize} \item 幂级数$\sum\limits_{n=0}^\infty a_nx^n$的和函数$S(n)$在其收敛区间$I$上连续,且如果幂级数在收敛区间的端点$x=\pm R$处收敛,则和函数$S(x)$在$(-R,R]$或$[-R,R)$上连续。 - \item 幂级数$\sum\limits_{n=0}^\infty a_nx^n$的和函数$S(x)$在其收敛域$I$上可积,且有逐项积分公式$\int_0^xS(t)\,\textrm{d}t=\int_0^x(\sum\limits_{n=0}^\infty a_nt^n)\,\textrm{d}t=\sum\limits_{n=0}^\infty a_n\int_0^xt^n\,\textrm{d}t=\sum\limits_{n=0}^\infty\dfrac{a_n}{n+1}x^{n+1}$($x\in I$),逐项积分后得到的幂级数和原级数有相同收敛半径,但是收敛域可能扩大。 - \item 幂级数$\sum\limits_{n=0}^\infty a_nx^n$的和函数$S(x)$在其收敛区间$(-R,R)$内可导,且有逐项求导公式$S'(x)=(\sum\limits_{n=0}^\infty a_nx^n)'=\sum\limits_{n=0}^\infty(a_nx^n)'=\sum\limits_{n=1}^\infty na_nx^{n-1}$($\vert x\vert