\subsubsection {}
Nech� $f$ je bodovou p��p. stejnom�rnou limitou posloupnosti p��p. �ady 
funkc� $f_{n}$ na intervalu. Rozhodn�te, zda z toho plyne:

jsou-li $f_{n}$ omezen�, je $f$ omezen�;

jsou-li $f_{n}$ spojit�, je $f$ spojit�;

maj�-li $f_{n}$ spojit� derivace, m� $f$ spojitou derivaci;

maj�-li $f_{n}$ primitivn� funkci, m� $f$ primitivn� funkci; 

 



\subsubsection {}
Vy�et�ete bodovou a stejnom�rnou konvergenci posloupnosti funkc�:

$\frac{n^2 x^2}{1+n^2 x^2}, \frac{x^n}{e^{n(x-1)}}, \sin^n x \cos x, x^n 
- x^{2n}, x^n - x^{n+1},$ $ \operatorname{arccotg} (e^{-\frac{x}{n}}), 
\frac{\operatorname{arctg} (nx)}{nx}, \frac{x}{n}, \frac{x^n}{e^{nx}}, 
x^n, \frac{nx}{1+n+x},$ $ \sqrt{x^2+\frac{1}{n^2}}, \frac{[nf(x)]}{n}, 
nx(1-x)^n, e^{nx - n}, \root n \of {1+x^n}, \frac{x^3 
\frac{1}{n}}{x^6+\frac{1}{n^2}}, e^{-\sqrt{|x^2-n^2|}}.$
 



\subsubsection {}
Vy�et�ete bodovou a stejnom�rnou konvergenci �ady funkc�:

$$
\sum\limits_{n=1}^{\infty} \frac{x^2}{1+n^2+n^2x^2}, 
\sum\limits_{n=1}^{\infty} (\frac{x^n}{n} - \frac{x^{n+1}}{n+1}),
\sum\limits_{n=1}^{\infty} \operatorname{arccotg} (n^x+n^{\frac{1}{x}}),
$$

$$
\sum\limits_{n=1}^{\infty} \operatorname{arctg}(\frac{x}{n^2}),
\sum\limits_{n=1}^{\infty} \frac{\sin nx}{\root 3 \of {n^4+x^4}},
$$

$$
\sum\limits_{n=2}^{\infty} \frac{(-1)^n x}{x+n},
\sum\limits_{n=2}^{\infty} \frac{n^2}{\sqrt{n!}}(x^n+\frac{1}{x^n}), 
\frac{1}{2}\le x \le 2.
$$

 



\subsubsection {}
Doka�te, �e $\sum\limits_{n=2}^{\infty} \frac{\sin nx}{n^3}$ m� spojitou 
derivaci.

\subsubsection {}
Vy�et�ete konvergenci (absolutn�,
neabsolutn�,
stejnom�rnou,
lok�ln� stej\-no\-m�rnou) �ady
$$\sum\limits_{n=1}^\infty
{\left(-1\right)^{n(n-1)\over 2}\over \root 3\of{n^2+e^x}}\qquad
(8\ b).$$




\subsubsection {}
Vy�et�ete konvergenci (absolutn�,
neabsolutn�,
stejnom�rnou,
lok�ln� stej\-no\-m�rnou) �ady
$$\sum\limits_{n=1}^\infty
{\cos{2n\pi\over 3}\over \root 3\of{n^2+e^x}}\qquad (9\ b).$$





