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%\section{ANALIZA I - septembar  2002.  I tok }
\qquad     \qquad ANALIZA I - septembar  2002.  \quad I tok  
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{\bf 1.} Dat je realan niz $a_n$: $a_0$ , $a_{n+1}=\frac{2a_n}{(n+2)a_n+1} $
za  $n=0,1,2,...$\\ a) Dokazati da je 
$$\frac{1}{n+\frac{q}{n}} \leq a_n \leq \frac{1}{n} $$ za neko $q > 0$ i  $n =1,2,..$ \\
b) Ispitati konvergenciju realnog reda $$\sum_{n=0}^\infty a_nx^n $$
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{\bf 2.} Ispitati funkciju $\sqrt{x^2 + 2x} e^{ \frac{ 1}{ x}}$ i nacrtati njen grafik.

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{\bf 3.} Izra\v{c}unati $$\int_0^1\frac{arcsin x - x}{x^3} \, dx$$
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{\bf 4.} Neka je $f:[0,1]  \rightarrow {\bf R} $ neprekidna funkcija, diferencijabilna na (0,1), f(0) = f(1) = 0, $ 0 < \alpha < \beta < 1 $
i $f( \alpha ) - f( \beta ) = 1$. Dokazati da je $ | f^ {\prime} ( \xi ) | = 2 $
za neko $0 < \xi < 1$.
  
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