\subsubsection {}
Vy�et�ete konvergenci n�sleduj�c�ch �ad 
(p��padn� pro kter� hodnoty parametr� konverguj�).
Vy�et�ete konvergenci absolutn� a neabsolutn�.

 
1 ${\sum\limits_{k=1}^\infty} {(k!)^2\over (2k)!}$ 

{2.} ${\sum\limits_{k=1}^\infty} {k!\over k^k}$

{3.} ${\sum\limits_{k=1}^\infty} {(k!)^2\over k^k}$

{4.} ${\sum\limits_{k=1}^\infty} \arcsin^k{k\over k\sqrt2+1}$

{5.} ${\sum\limits_{k=1}^\infty} (-1)^k
\left( {2k+100\over 3k+1} \right) ^k $

{6.} ${\sum\limits_{k=1}^\infty} \sin k$

7. ${\sum\limits_{k=1}^\infty} k^{\ln x}, \quad x\in\Bbb R$\quad 

{8.} $ {\sum\limits_{k=2}^\infty} 
{\sqrt{k+2}-\sqrt{k-2}\over
k^\alpha},\quad \alpha\in\Bbb R $
\quad

{9.} {\sum\limits_{k=1}^\infty} \tg{k^\alpha+1\over k^\beta +1},\quad 0\le\alpha\le\beta
$

10.  ${\sum\limits_{k=1}^\infty} {x^k\over 1+x^{2k}},\quad x\in\Bbb R %$
\quad
 
{11.} ${\sum\limits_{k=1}^\infty} {(k-1)! e^k\over k^k}{x^k\over 1+x^{2k}},\quad
x\in\Bbb R$\quad

{12.}$ {\sum\limits_{k=0}^\infty} k x e^{-kx} \cos kx,\quad x\in\Bbb R$
%

{13.} ${\sum\limits_{k=1}^\infty} \left( 1+\frac 1k \right) ^{k^2} x^k,\quad x\in\Bbb R$
\quad 

{14.} $ {\sum\limits_{k=2}^\infty}  \sin^2 {x\over \sqrt k \ln k},\quad x\in\Bbb R$
%
\quad

{15.} ${\sum\limits_{k=0}^\infty} {z^{k^2}\over 2^k},\quad z\in\Bbb C$

{16.} ${\sum\limits_{k=1}^\infty} \sin\left( \pi\sqrt{k^2+1}\right) \root k \of {x^2\over
x^2+1}, \quad x\in\Bbb R$


{17.} ${\sum\limits_{k=1}^\infty} (-1)^k{k\sin k\over k^2+1}$ \quad  

{18.} ${\sum\limits_{k=1}^\infty}
(-1)^k{k\sin^2 k\over k^2+1}$ \quad 

{19.} $ {\sum\limits_{k=2}^\infty}  (-1)^k{\root k\of k
\over \ln k}$ 

{20.} ${\sum\limits_{k=1}^\infty} (-1)^k {\sqrt k\over k+1} {\sinh k^2 +\cosh k^2 \over
\cosh k^2 +1}\quad {21.} {\sum\limits_{k=1}^\infty} {z^k\over k+\sqrt k},\quad z\in\Bbb C$



\subsubsection {}
Zkoumejte konvergenci (absolutn� a neabsolutn�)

1. ${\sum\limits_{k=1}^\infty}{(3+4i)^k\over a^k k^2}$\quad 

2. ${\sum\limits_{k=1}^\infty}\left({1+\cos k\over
2+\cos k}\right)^{2k-\ln k}$\quad 

3. ${\sum\limits_{k=2}^\infty} {(-1)^k\over \sqrt k+(-1)^k}$

4. ${\sum\limits_{k=2}^\infty} {(-1)^k\over  k+(-1)^k}$

 5. ${\sum\limits_{k=2}^\infty} {(-1)^k\over 2k+(-1)^k}$\quad 

6. ${\sum\limits_{k=1}^\infty}{(-1)^{[\sqrt
k]}\over k}$\quad 

7. ${\sum\limits_{k=1}^\infty}\sin\left(\pi\sqrt{k^2+k}\right)$ \quad 

8. ${\sum\limits_{k=1}^\infty}{\sin
kx\over k}$

9.  Nech� ${\sum\limits_{k=1}^\infty} (a_k)^2<\infty$. Doka�te, �e ${\sum\limits_{k=1}^\infty}{a_k\over k}$
konverguje absolutn�.


