Součásti dokumentu Matematika2Priklady
Zdrojový kód
%\wikiskriptum{Matematika2Priklady}
\section{Mocninné řady}
\subsection{Obor konvergence mocninných řad.}
\begin{enumerate}
\begin{priklad}
\sum n x^n; M = (-1, 1)
\end{priklad}
\begin{priklad}
\sum \frac{1}{(2n)!}x^n; M = \cal{R}
\end{priklad}
\begin{priklad}
\sum (-n)^{2n} x^{2n}; M = \{ 0 \}
\end{priklad}
\begin{priklad}
\sum \frac{1}{n2^n}x^n; M = \langle -2, 2 )
\end{priklad}
\begin{priklad}
\sum \left ( \frac{n}{100} \right ) ^n x^n; M = \{ 0 \}
\end{priklad}
\begin{priklad}
\sum \frac{2^n}{\sqrt{n}}x^n ; M = \left \langle -\frac{1}{2}, \frac{1}{2}\right )
\end{priklad}
\begin{priklad}
\sum \frac{n-1}{n} x^n; M = (-1, 1)
\end{priklad}
\begin{priklad}
\sum \frac{n}{10^n}x^n; M = (-10, 10)
\end{priklad}
\begin{priklad}
\sum \frac{x^n}{n^n}; M = \cal{R}
\end{priklad}
\begin{priklad}
\sum \frac{(-1)^n}{n^n}(x-2)^n; M = \cal{R}
\end{priklad}
\begin{priklad}
\sum \frac{\ln n}{2^n}(x-2)^n; M = (0, 4)
\end{priklad}
\begin{priklad}
\sum (-1)^n \left ( \frac{2}{3}\right) ^n (x+1)^n; M = \left (
-\frac{5}{2}, \frac{1}{2} \right )
\end{priklad}
\begin{priklad}
\sum \frac{5^n}{n}(x-2)^n; M = \left \langle \frac{9}{5}, \frac{11}{5} \right )
\end{priklad}
\begin{priklad}
\sum n(n+1)(x-1)^{2n}; M = (0, 2)
\end{priklad}
\begin{priklad}
\sum \frac{n}{2n+1} x^{2n+1}; M = (-1, 1)
\end{priklad}
\begin{priklad}
\sum \frac{n!}{2}(x+1)^n; M = \{ -1\}
\end{priklad}
\begin{priklad}
\sum \frac{(-1)^nn}{3^{2n}}x^n; M = (-9, 9)
\end{priklad}
\begin{priklad}
\sum \frac{(-1)^n}{5^{n=1}}(x-2)^n; M = (-3, 7)
\end{priklad}
\end{enumerate}
\separator
\subsection{Rozvoj funkce do mocninné řady}
\begin{enumerate}
\begin{priklad}
\frac{1}{(1-x)^k} = 1 + kx + \frac{(k+1)k}{2!}x^2 +
\sum_{n=3}^{+\infty} \frac{(n+k-1)!}{n!(k-1)!}x^n
\end{priklad}
\begin{priklad}
\ln(1-x^2) = \sum (-1)^{n+1} \frac{x^{2n+2}}{n+1}
\end{priklad}
\begin{priklad}
x^2 \sin x
\end{priklad}
\begin{priklad}
\sin x^2 = \sum_{n=0}^{+\infty} \frac{(-1)^n}{(2n+1)!}x^{4n+2}
\end{priklad}
\begin{priklad}
e^{3x^3} = \sum_{n=0}^{+\infty} \frac{3^n}{n!}x^{3n}
\end{priklad}
\begin{priklad}
\frac{2x}{1-x^2} = 2 \sum_{n=0}^{+\infty}x^{2n+1}
\end{priklad}
\begin{priklad}
\frac{1}{1-x} + e ^x = \sum_{n=0}^{+\infty} \frac{n!+1}{n!}x^n
\end{priklad}
\begin{priklad}
x \ln(1+x^3) = \sum_{n=1}^{+\infty} \frac{(-1)^{n+1}}{n} x ^{3n+1}
\end{priklad}
\begin{priklad}
x^3 e ^{-x^3} = \sum_{n=0}^{+\infty} \frac{(-1)^n}{n!}x^{3n+3}
\end{priklad}
\begin{priklad}
\sqrt{1-x^2}
\end{priklad}
\begin{priklad}
\frac{1}{\sqrt{1+x}}
\end{priklad}
\begin{priklad}
\frac{1}{\sqrt[3]{1+x}}
\end{priklad}
\begin{priklad}
\sqrt[4]{1-x}
\end{priklad}
\begin{priklad}
x e^{5x^2} = \sum_{n=0}^{+\infty} \frac{5^n}{n!}x^{2n+1}
\end{priklad}
\begin{priklad}
\sqrt{x} \arctg{\sqrt{x}} = \sum \frac{(-1)^n}{2n+1}x^{n+1}
\end{priklad}
\begin{priklad}
(x+x^2)\sin{x^2} = \sum_{n=0}^{+\infty}
\frac{(-1)^n}{(2n+1)!}(x^{4n+3} + x^{4n+4})
\end{priklad}
\begin{priklad}
x \arctg{x} - \ln {x^2+1} = \sum_{n=1}^{+\infty}
\frac{(-1)^n}{2n(2n-1)}x^{2n}; M = \langle -1, 1 \rangle
\end{priklad}
\begin{priklad}
\arctg x \left ( \frac{2-2x}{1+4x} \right ) = \arctg x +
\sum_{n=0}^{+\infty} \frac{(-1)^{n+1}}{2n+1}(2x)^{2n+1}; M =
\left \langle - \frac{1}{4}, \frac{1}{2} \right \rangle
\end{priklad}
\begin{priklad}
\frac{1}{4} \ln \left( \frac{1+x}{1-x} \right ) + \frac{1}{2}
\arctg x = \sum_{n=0}^{+\infty} \frac{x^{4n+1}}{4n+1}; M = (-1, 1)
\end{priklad}
\end{enumerate}
\separator
\subsection{Sčítání řad}
\begin{enumerate}
\begin{priklad}
\sum_{n=0}^{+\infty} x^{5n+1} = \frac{x}{1-x^5}
\end{priklad}
\begin{priklad}
\sum_{n=0}^{+\infty} 2 x^{3n+2}
\end{priklad}
\begin{priklad}
\sum_{n=1}^{+\infty} \frac{3}{2}x^{2n-1} \frac{3x}{2(1-x^2)}
\end{priklad}
\begin{priklad}
\sum_{n=1}^{+\infty} \frac{x^n}{(n-1)!}
\end{priklad}
\begin{priklad}
\sum_{n=1}^{+\infty} \frac{n^2}{n!}= 2e
\end{priklad}
\end{enumerate}