%% This document created by Scientific Notebook (R) Version 3.5 %% Starting shell: article \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amssymb} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{Created=Wednesday, February 10, 1999 13:29:48} %TCIDATA{LastRevised=Sunday, February 13, 2005 16:29:58} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{Counters=arabic,1} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za I online - Zbierka 5\dotfill \thepage }} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \author{A. U. Thor} \title{Lab Report} \date{The Date } \maketitle \begin{abstract} A Laboratory report created with Scientific Notebook \end{abstract} \section{Zbierka \'{u}loh} \begin{center} \begin{tabular}{|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Zbierka}{}{}{Z.tex}}% %BeginExpansion \msihyperref{Zbierka}{}{}{Z.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \subsection{Postupnosti.} \textbf{Pr\'{\i}klad 1. }Nap\'{\i}\v{s}te prv\'{y}ch p\"{a}\v{t} prvkov postupnosti $\left\{ \left( \frac{n+5}{n}\right) ^{n}\right\} _{n=1}^{\infty }\ $a vypo\v{c}\'{\i}tajte (ak existuje) jej limitu. $% \CustomNote{Answer}{$6,\,\left( \frac{7}{2}\right) ^{2},\,\left( \frac{8}{3}% \right) ^{3},\,\left( \frac{9}{4}\right) ^{4},\,\left( \frac{10}{5}\right) ^{5},\;e.$}$ \textbf{Pr\'{\i}klad 2. }Nap\'{\i}\v{s}te prv\'{y}ch p\"{a}\v{t} prvkov postupnosti $\left\{ \left( \frac{n+6}{n+5}\right) ^{n}\right\} _{n=1}^{\infty }\ $a vypo\v{c}\'{\i}tajte (ak existuje) jej limitu. $% \CustomNote{Answer}{$\frac{7}{6},\,\left( \frac{8}{7}\right) ^{2},\,\left( \frac{9}{8}\right) ^{3},\,\left( \frac{10}{9}\right) ^{4},\,\left( \frac{11}{% 10}\right) ^{5}\,,\,e.$}$ \textbf{Pr\'{\i}klad 3. }N\'{a}jdite n-t\'{y} \v{c}len postupnosti $\left\{ 0,9;0,99;0,999;\dots \right\} $ a vypo\v{c}\'{\i}tajte jej limitu. $% \CustomNote{Answer}{$a_{n}=1-\left( \frac{1}{10}\right) ^{n},\,\lim_{n\longrightarrow \infty }a_{n}=1.$}$ \textbf{Pr\'{\i}klad 4. }N\'{a}jdite infimum, supremum, minimum, maximum postupnosti $\left\{ \frac{2n+1}{n+2}\right\} _{n=1}^{\infty }.$ $% \CustomNote{Answer}{$\inf a_{n}=\min a_{n}=1,\,\sup a_{n}=2,\,\max a_{n}\,\nexists .$}$ \textbf{Pr\'{\i}klad 5. }Zistite, \v{c}i je postupnos\v{t} $\left\{ \frac{2n% }{n+3}\right\} _{n=1}^{\infty }$\ a. monot\'{o}nna, b. ohrani\v{c}en\'{a}, c. konvergentn\'{a}. $% \CustomNote{Answer}{% a. rast\'{u}ca, b. ohrani\v{c}en\'{a}, c. $\lim_{n\longrightarrow \infty }a_{n}=2.$}$ \textbf{Pr\'{\i}klad 6. }Zistite, \v{c}i je postupnos\v{t} $\left\{ \frac{n+1% }{2n}\right\} _{n=1}^{\infty }$\ \ a.\ monot\'{o}nna, b. ohrani\v{c}en\'{a}, c. konvergentn\'{a}. $% \CustomNote{Answer}{% a. klesaj\'{u}ca, b. ohrani\v{c}en\'{a}, c. $\lim_{n\longrightarrow \infty }a_{n}=\frac{1}{2}.$}$ \textbf{Pr\'{\i}klad 7. }Zistite, \v{c}i je postupnos\v{t} $\left\{ \frac{% 1+2+3+\dots +n}{n+2}-\frac{n}{2}\right\} _{n=1}^{\infty }$\ a. monot\'{o}nna, b. konvergentn\'{a}. $% \CustomNote{Answer}{% a. klesaj\'{u}ca, b. $\lim_{n\longrightarrow \infty }a_{n}=-\frac{1}{2}.$}$ \textbf{Pr\'{\i}klad 8. }Pomocou defin\'{\i}cie limity postupnosti dok\'{a}% \v{z}te, \v{z}e $\lim_{n\longrightarrow \infty }\frac{n-1}{n+1}=1,$\ n\'{a}% jdite pr\'{\i}slu\v{s}n\'{e} $n_{0},$ ak $\varepsilon =\frac{1}{10}.$ $% \CustomNote{Answer}{$n_{0}=19.$}$ \textbf{Pr\'{\i}klad 9. }Pomocou defin\'{\i}cie limity postupnosti dok\'{a}% \v{z}te, \v{z}e $\lim_{n\longrightarrow \infty }\frac{2^{n}+1}{2^{n}}=1,$\ n% \'{a}jdite pr\'{\i}slu\v{s}n\'{e} $n_{0},$ ak $\varepsilon =\frac{2}{100}.$ $% \CustomNote{Answer}{$n_{0}=6.$}$ \textbf{Pr\'{\i}klad 10. }Zistite, \v{c}i je postupnos\v{t} $\left\{ \frac{% 3^{n}+3}{1-4\cdot 2^{n}}\right\} _{n=1}^{\infty }$\ konvergentn\'{a}. \CustomNote{Answer}{% divergentn\'{a}} \textbf{Pr\'{\i}klad 11. }Zistite, \v{c}i s\'{u} postupnosti konvergentn\'{e} \ a. $\left\{ \left( 3-\frac{1}{n}\right) \sqrt{\frac{n}{4n+1}}\right\} _{n=1}^{\infty },$ b. $\left\{ \left( \frac{1}{2}+\frac{1}{4n}\right) ^{2n}\right\} _{n=1}^{\infty }.$ \CustomNote{Answer}{% a. konvergentn\'{a}, b. konvergentn\'{a}} \textbf{Pr\'{\i}klad 12. }Zistite, \v{c}i s\'{u} postupnosti konvergentn\'{e} \ a. $\left\{ 1+\cos \left( n\pi \right) \right\} _{n=1}^{\infty },$ b. $\left\{ \frac{1-2+3-4+5+\dots +\left( -1\right) ^{n-1}n}{n}\right\} _{n=1}^{\infty }.$ \CustomNote{Answer}{% a. divergentn\'{a}, b. konvergentn\'{a}} \textbf{Pr\'{\i}klad 13. }Vy\v{s}etrite $\lim_{n\longrightarrow \infty }% \frac{a^{n}}{1+a^{n}},$ ak $a>0.$ $% \CustomNote{Answer}{$% \begin{array}{c} \text{ak }01\Longrightarrow \lim_{n\longrightarrow \infty }\frac{a^{n}}{% 1+a^{n}}=1.% \end{array}% $}$ V pr\'{\i}kladoch 14 - 18 vypo\v{c}\'{\i}tajte limitu postupnosti, ak: \textbf{Pr\'{\i}klad 14. } \textbf{a. }$a_{n}=\sqrt{1+n^{2}}-n.$ $% \CustomNote{Answer}{$0.$}$ b. $a_{n}=\sqrt{n}\left( \sqrt{n+1}-\sqrt{n}\right) .$ $% \CustomNote{Answer}{$\frac{1}{2}.$}$ c. $a_{n}=\frac{1}{n^{2}}+\frac{2}{n^{2}}+\frac{3}{n^{2}}+\dots +\frac{n}{% n^{2}}.$ $% \CustomNote{Answer}{$\frac{1}{2}.$}$ d. $a_{n}=\left( 1+\frac{1}{4n}\right) ^{1-3n}.$ $% \CustomNote{Answer}{$e^{-\frac{3}{4}}.$}$ \textbf{Pr\'{\i}klad 15. }$a_{n}=\frac{1+3+5+\dots +\left( 2n-1\right) }{n+1}% -\frac{2n+1}{2}.$ $% \CustomNote{Answer}{$-\frac{3}{2}.$}$ \textbf{Pr\'{\i}klad 16. }$a_{n}=1-\frac{1}{3}+\frac{1}{9}-\dots +\frac{% \left( -1\right) ^{n-1}}{3^{n-1}}.% \CustomNote{Answer}{$\frac{3}{4}.$}$ \textbf{Pr\'{\i}klad 17. }$a_{n}=\left( \frac{3n-2}{3n+1}\right) ^{5n-3}.$ $% \CustomNote{Answer}{$e^{-5}.$}$ \textbf{Pr\'{\i}klad 18. }$a_{n}=\left( \frac{n+1}{n-2}\right) ^{2n-1}.$ $% \CustomNote{Answer}{$e^{6}.$}$ \textbf{Pr\'{\i}klad 19. }Dan\'{a} geometrick\'{a} postupnos\v{t} m\'{a} kvocient $q=2$ a s\'{u}\v{c}et jej prv\'{y}ch piatich prvkov je $s_{5}=124.$ Vypo\v{c}\'{\i}tajte $a_{5}$\ a n\'{a}jdite s\'{u}\v{c}et geometrickej postupnosti, ak existuje. $% \CustomNote{Answer}{$a_{5}=64,\,s$ neexistuje.}$ \textbf{Pr\'{\i}klad 20. }Zistite, \v{c}i je postupnos\v{t}$\left\{ \frac{% 4^{n}\left( n!\right) ^{2}}{\left( 2n\right) !}\right\} _{n=1}^{\infty }$\ monot\'{o}nna. \v{C}o viete o konvergencii radu $\sum_{n=1}^{\infty }\frac{% 4^{n}\left( n!\right) ^{2}}{\left( 2n\right) !}.$ \CustomNote{Answer}{% postupnos\v{t} je rast\'{u}ca, \par $\lim_{n\rightarrow \infty }\frac{4^{n}\left( n!\right) ^{2}}{\left( 2n\right) !}=\allowbreak \infty \neq 0,$\ \par rad nemo\v{z}e by\v{t} konvergentn\'{y} (nutn\'{a} podmienka nie je splnen% \'{a})} \begin{center} \begin{tabular}{|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Zbierka}{}{}{Z.tex}}% %BeginExpansion \msihyperref{Zbierka}{}{}{Z.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za I} \section{Zbierka \'{u}loh} \end{document}