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\NoBlackBoxes
\centerline{\bf Èíselné øady}

\medskip
Pøíklad 1:

Zjistìte, zda konvergují resp divergují øady 
$$
\align \split
&\sum_{n=1}^{\infty}(-1)^n \frac{2n^2 + 3n + 4}{2n^2 + 5}, \sum_{n=1}^{\infty} \frac{2n^2 + 3n + 4}{(2n^2 + 5)^2}, \sum_{n=1}^{\infty} \frac{n^2}{n^3 + 1},\\
& \sum_{n=1}^{\infty}\frac{2^n}{n!}, \sum_{n=1}^{\infty} \frac{(n!)^2}{(2n)!}, \sum_{n=1}^{\infty}\frac{n!}{n^n}.
\endsplit
\endalign
$$

\medskip
Pøíklad 2:

Zjistìte, zda konvergují resp divergují øady 
$$
\align \split
&\sum_{n=1}^{\infty}\frac{\root 3 \of{n^2 + 5} - \root 3 \of{n^2 + 1}}{\root 4 \of n}, \sum_{n=1}^{\infty}\frac{1}{\sqrt{2n+1}\sqrt{2n+3}}, \sum_{n=1}^{\infty}\frac{n^5}{2^n + 3^n}.
\endsplit
\endalign
$$
\medskip
Pøíklad 3:

Zjistìte, zda konvergují øady $(a,b > 0)$
$$\align \split
&\sum_{n=1}^{\infty} \frac{a \cos^2 \frac{4n\pi}{9}}{2^n},\ \ 
\sum_{n=1}^{\infty} \frac{n!}{n^{n+p}},\ \ \sum_{n=1}^{\infty}(\sqrt{2}- 
\root 3\of{2})(\sqrt{2} - \root 5 \of{2})...(\sqrt{2}- 
\root{2n+1}\of{2}),\\  &\sum_{n=1}^{\infty} 
\frac{n^{2n}}{(n+a)^{n+b}(n+b)^{n+a}}.\endsplit \endalign
$$

\medskip
Pøíklad 4:

Pro která $\alpha$ konverguje øada $\sum_{n=1}^{\infty} 
\frac{\sqrt{n+2} - \sqrt{n-2}}{n^{\alpha}}$?


\medskip
Pøíklad 5:
Využijte limit 
$$
 \lim_{y\to 0}\frac{\sin y }{y }= 1, \lim_{y\to 0} \frac{1 - \cos 
 y}{y^2} = \frac12, \lim_{y\to 0}\frac{\ln (1+y)}{y} = 1
$$ 
a zjistìte, pro která $\alpha$ konvergují øady
$$
\sum_{n=1}^{\infty} (1 - \cos \frac1n)n^3,\ \ \sum_{n=1}^{\infty}(1 - 
\cos \frac1n)^{\alpha} \ln n,\ \  \sum_{n=1}^{\infty}\sin\frac1n \ln 
\frac{n+1}{n}.
$$

\medskip

Pøíklad 6:

Zjistìte, zda konvergují resp. absolutnì konvergují øady
$$
\sum_{n=1}^{\infty}(\cos n^2)(\sqrt{n^6 + 6n} - n^3), 
\sum_{n=1}^{\infty}\cos (n^2(\sqrt{n^6 + 6n} - n^3)), 
\sum_{n=1}^{\infty}\frac{(-3)^{n+1}- 3^{n+1}}{n}.
$$


\medskip
Pøíklad 7:

Zjistìte, pro která $x \in R$ konvergují (resp. absolutnì konvergují) øady
$$
\sum_{n=1}^{\infty}n^3 x^n, \sum_{n=1}^{\infty}\frac{2^n}{n!}x^n, 
\sum_{n=1}^{\infty}\frac{2^n}{n^2}x^n, 
\sum_{n=1}^{\infty}\frac{n^3}{3^n}x^n, 
\sum_{n=1}^{\infty}\frac{(-1)^{n+1}x^n}{n}, 
\sum_{n=1}^{\infty}\frac{x^n}{n^2}, 
\sum_{n=1}^{\infty}\frac{(-1)^{n}x^{2n+1}}{2n+1}. 
$$

\medskip
Pøíklad 8:

Zjistìte, zda  konvergují øady
$$
\sum_{n=1}^{\infty}(-1)^n (\root n \of{3}- 1), 
\sum_{n=1}^{\infty}\frac{\sin \frac{n\pi}{3}}{\sqrt{n+1}}, 
\sum_{n=1}^{\infty}\frac{(-1)^n}{n + (-1)^n}. 
$$
\medskip
Pøíklad 9:
Vyšetøete konvergenci øad
$$\sum_{n=0}^{\infty}(-1)^n{2n^2+3n+4\over 2n^2+5},
\sum_{n=20}^{\infty}(-1)^n{2n^2+3n+4\over (2n^2+5)^2},
\sum_{n=2}^{\infty}(-1)^n({n-1\over n+1})^n,$$
$$\sum_{n=0}^{\infty}{n^3\over n^3+1},
\sum_{n=1}^{\infty}{n^2\over (2+{1\over n})^n},
\sum_{n=0}^{\infty}{n^5\over 2^n+3^n},$$
$$\sum_{n=1}^{\infty}{n\over 2^n},
\sum_{n=0}^{\infty}{1\over \sqrt{2n+1}\sqrt{2n+3}},
\sum_{n=1}^{\infty}{(n!)^2\over 2^{n^2}},
\sum_{n=1}^{\infty}{n^{n+{1\over n}}\over n+{1\over n}}.$$
\medskip
Pøíklad 10:
Rozhodnìte, pro která reálná $x$ konverguje øada
$$\sum_{n=1}^{\infty}{(-1)^n \over (2n-1)}(\frac{1-x}{1+x})^n.$$


\bye
