From 3ea9cd8401ac9dbc521bbaa734d73d1c5222dc4f Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Fri, 25 Jun 2021 23:39:11 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E5=AE=9A=E7=A7=AF=E5=88=86?= =?UTF-8?q?=E9=83=A8=E5=88=86?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...efinite-integral-and-definite-integral.pdf | Bin 129977 -> 136279 bytes ...efinite-integral-and-definite-integral.tex | 22 ++++++++++++ ...efinite-integral-and-definite-integral.pdf | Bin 176373 -> 180468 bytes ...efinite-integral-and-definite-integral.tex | 32 +++++++++++++++++- 4 files changed, 53 insertions(+), 1 deletion(-) diff --git a/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf b/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf index 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a/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex +++ b/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex @@ -451,6 +451,28 @@ $=-\dfrac{4x+3}{2(x^2+x+1)}-\dfrac{6}{\sqrt{3}}\arctan\dfrac{2x+1}{\sqrt{3}}+C$ \section{定积分} +\subsection{定限积分与极限} + +若极限中有$n$这种变量,也可以通过定积分的定义来做,$\lim\limits_{n\to\infty}\sum\limits_{i=1}^nf\left(\dfrac{i}{n}\right)\dfrac{1}{n}=\int_1^0f(x)\,\textrm{d}x$。 + +\begin{enumerate} + \item 先提出$\dfrac{1}{n}$。 + \item 凑出$\dfrac{i}{n}$。 + \item 写出$\int_0^1f(x)\,\textrm{d}x$。 +\end{enumerate} + +\textbf{例题:}求$\lim\limits_{n\to\infty}\left(\dfrac{1}{n+1}+\dfrac{1}{n+2}\cdots+\dfrac{1}{n+n}\right)$。 + +即求$\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{1}{n+i}$。如果我们要传统求的话一般使用夹逼准则,找到放缩的两个函数。 + +所以找到两个:$\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{1}{n+n}<\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{1}{n+i}<\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{1}{n+1}$。 + +即$\dfrac{1}{2}<\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{1}{n+i}<\lim\limits_{n\to\infty}\dfrac{n}{n+1}=1$。夹逼准则失败。 + +所以对$\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{1}{n+i}$通过定积分定义进行计算。 + +$\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{1}{n+i}=\lim\limits_{n\to\infty}\sum\limits_{i=1}^n\dfrac{1}{n+\dfrac{i}{n}}\dfrac{1}{n}=\displaystyle{\int_0^1\dfrac{1}{1+x}\,\textrm{d}x}=[\ln(1+x)]_0^1=\ln2$。 + \subsection{变限积分} \subsection{牛莱公式} diff --git a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf index f248e44eaae6b44613aad2e46846244420eec558..2b30b3c681055ba094845239de2ddb2f26d93e2e 100644 GIT binary patch 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