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\Nadpis{\bf 5. cvi�en� z MA --- 3. a 5.11.2008}

(na 4. cvi�en� jsme po��tali p��klady z pap�ru ke t�et�mu cvi�en�)

\nadpis{Dal�� limity}

\pr $\lp {n+\floor{\root 3 \of n}^3\over n - \floor{\sqrt{n+9}} }$
\myskip
\pr $\lp\frac in \left({1+i\over\sqrt 2}\right)^n$
\myskip
\pr $\lp \frac { 3^n + 2n^n + n!}{ (n+1)^4 + \sin n + (3n)! }$

\pr 
\cast $\lp\sqrt n(\root n\of 3-\root n \of 2)$
\cast $\lp(\root 3\of{\sqrt {n^7} +\root 3 \of {n^7}} 
    -\root 3\of{\sqrt {n^7}-\root 3\of {n^7}})$

\pr Spo�t�te v z�vislosti na $k,l\in\Bbb N$
\cast $\lp{n^k-(n-1)^l\over n^k+n^l}$;
\cast $\lp{(n+1)^k+(-n)^l\over(n-1)^k-n^l}$.

\pr \cast $\l (1+1/n)^{n^2}$, \cast $\l (1+1/n^2)^{n}$

\pr Doka�te, �e sou�in 
 $\frac21\cdot\frac23\cdot\frac43\cdot\frac 45\cdot\frac65\cdot\dots$ m� kone�nou nenulovou hodnotu.

\pr �emu se rovn� $\prod\limits_{n=1}^\infty\left(1-{1\over (n+1)^2}\right)$?

\pr * Doka�te, �e $\lp\sum\limits_{i=0}^k a_i\sqrt{n+i}=0$,
pokud $\sum\limits_{i=0}^k a_i=0$.

\pr * Pro kter� $x\in\Bbb R$ existuje $\lp\sin nx$ ? Tot� pro $\lp\cos nx$.


\oddel

\pr Rekurentn� zadan� posloupnosti --- vypo�t�te $\l a_n$ (a nezapome�te 
si rozmyslet, zda limita v�bec existuje) pro
 \cast $a_1 = t$ ($t>0$ je parametr), $a_{n+1} = (a_n + 2/a_n)/2$,
 \cast $a_1=\sqrt 2$, $a_{n+1} = \sqrt {2+a_n}$,

 \cast $a_1=1$, $a_{n+1} = 1/(1+a_n)$.
 \cast $a_1=\sqrt2,\,a_{n+1}=\sqrt{2-a_n}$

 \cast $a_1=0$, $a_{n+1} = a_n + (x-a_n)^2/2$ ($0 \le x \le 1$)
  
 \cast (Jako varov�n�!) $a_1=0.4$, $a_{n+1} = 4a_n (1-a_n)$.


\oddel

\pr
Rozhodn�te o platnosti n�sleduj�c�ch tvrzen�:

\cast Pokud $\l a_n = +\infty$, potom $\l 1/a_n = 0$.
\cast Pokud $\l a_n = 0$, potom $\l 1/a_n = +\infty$.
\cast $\l a_n = 0$, pr�v� kdy� $\l 1/|a_n| = +\infty$.
\cast Pokud $\l a_n = A$, potom $\l a_{2n} = A$.
\cast Pokud $\l a_{2n} = A$, potom $\l a_{n} = A$.
\cast $\l a_n = A$, pr�v� kdy� $\l a_{b_n} = A$ pro ka�dou
 rostouc� posloupnost p�irozen�ch ��sel $b_n$.
\cast $\l a_n = A$, pr�v� kdy� $\l a_{b_n} = A$ pro ka�dou
 neklesaj�c� posloupnost p�irozen�ch ��sel $b_n$.



\pr
Co je to limes superior/inferior? Jak souvis� s limitou, se supremem/infimem? 
Ur�ete limes superior i inferior posloupnost� 
$(n-1)/n$,
$(-1)^n$, $(-1)^n-1/n$, 
$5^{(-1)^n 3^n}$. 


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