<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%% 
%%  Ein Beispiel der DANTE-Edition
%%  Mathematiksatz mit LaTeX
%%  3. Auflage
%%  Beispiel 04-20-7 auf Seite 76.
%%  Copyright (C) 2018 Herbert Voss
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$\begin{array}{&gt;{\displaystyle}r@{\kern1.4pt}&gt;{\displaystyle}l}
A=\lim _{n\rightarrow\infty}U &amp;= \lim_{n\rightarrow\infty}\sum^{n-1}_{i=0}\left(\Delta x\cdot f(a+i\cdot\Delta x\right) \\
 &amp;= \lim_{n\rightarrow\infty}\left(\Delta x\cdot f(a)+\Delta x\cdot f(a+\Delta x)+\Delta x\cdot f(a+2\cdot\Delta x)+\Delta x\cdot f(a+3\cdot \Delta x)+\ldots +\Delta x\cdot f(a+(n-1)\cdot \Delta x)\right) \\
 &amp;= \lim_{n\rightarrow\infty}\Delta x\cdot\left(f(a)+f(a+\Delta x)+f(a+2\cdot \Delta x)+f(a+3\cdot\Delta x)+\ldots+f(a+(n-1)\cdot\Delta x)\right) \\
 &amp;= \lim_{n\rightarrow\infty}\Delta x\left(a^2+\left(a+\Delta x\right)^2+\left(a+2\cdot\Delta x\right)^2+\left( a+3\cdot \Delta x\right)^2+\ldots +\left(a+(n-1)\cdot \Delta x\right)^2\right) \\
 &amp;= \lim_{n\rightarrow\infty}\Delta x\left(a^2+\left(a^2+2a\Delta x+\left(\Delta x\right)^2\right)+\left(a^2+2\cdot2a\Delta x+2^2\left(\Delta x\right)^2\right)+\left(a^2+2\cdot3a\Delta x+3^2\left( \Delta x\right)^2\right)+\ldots\right.\\
 &amp;  \qquad\left.\quad+\left(a^2+2\cdot(n-1)a\Delta x+(n-1)^2\left(\Delta x\right)^2\right)\right) \\
 &amp;= \lim_{n\rightarrow\infty}\Delta x\left(na^2+2a\Delta x\left( 1+2+3+\ldots+(n-1)\right)+\left(\Delta x\right)^2\left(1^2+2^2+3^2+\ldots+(n-1)^2\right)\right) \\
 &amp;= \lim_{n\rightarrow\infty}\Delta x\left(na^2+2a\Delta x\frac{n(n-1)}2+\left(\Delta x\right)^2\frac{n(2n-1)(n-1)}{6}\right) \\
 &amp;= \lim_{n\rightarrow\infty}\frac{b-a}{n}\left(na^2+2a\frac{b-a}{n}\frac{n(n-1)}2+\left(\frac{b-a}{n}\right)^2\frac{n(2n-1)(n-1)}{6}\right) \\
 &amp;= \lim_{n\rightarrow\infty}\frac{b-a}{n}\left(na^2+a(b-a)(n-1)+\frac{(b-a)^2}{n}\frac{(2n-1)(n-1)}{6}\right) \\
 &amp;= \lim_{n\rightarrow\infty}(b-a)\left(a^2+a(b-a)\frac{n-1}{n}+\frac{(b-a)^2}{n^2}\frac{(2n-1)(n-1)}{6}\right) \\
 &amp;= \lim_{n\rightarrow\infty}(b-a)\left(a^2+a(b-a)\left(1-\underbrace{\frac{1}{n}}\right)+(b-a)^2\frac{1}{6}\left(2-\underbrace{\frac{3}{n}}+\underbrace{\frac{1}{n^2}}\right)\right) \\
 \multicolumn{2}{r@{}}{\textrm{jeweils Nullfolgen f��r }n\rightarrow\infty}
\end{array}$}
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