[latex]a_{n}=frac{2n-2}{2n+1}\ a_{n+1}=frac{2(n+1)-2}{2(n+1)+1}=frac{2n}{2n+3}\ a_{n+1}-a_{n}=frac{2n}{2n+3}-frac{2n-2}{2n+1}=frac{2n(2n+1)}{(2n+3)(2n+1)}-frac{(2n-2)(2n+3)}{(2n+3)(2n+1)}=\ frac{4n^2+2n}{(2n+3)(2n+1)}-frac{4n^2-4n+6n-6}{(2n+3)(2n+1)}=frac{4n^2+2n}{(2n+3)(2n+1)}-frac{4n^2+2n-6}{(2n+3)(2n+1)}=\ =frac{4n^2+2n-(4n^2+2n-6)}{(2n+3)(2n+1)}=frac{4n^2+2n-4n^2-2n+6}{(2n+3)(2n+1)}=frac{6}{(2n+3)(2n+1)} extgreater 0[/latex]
[latex]\a_{n+1}= frac{2(n+1)-2}{2(n+1)+1} = frac{2n+2-2}{2n+2+1} = frac{2n}{2n+3} \ \a_{n+1}-a_n= frac{2n}{2n+3} - frac{2n-2}{2n+1} = frac{2n(2n+1)-(2n-2)(2n+3)}{(2n+3)(2n+1)} = \ \ frac{4n^2+2n-(4n^2+6n-4n-6)}{(2n+3)(2n+1)} = frac{4n^2+2n-4n^2-2n+6}{(2n+3)(2n+1)} = \ \ frac{6}{(2n+3)(2n+1)} extgreater 0implies ciag {a_n} rosnacy, \ \co nalezalo wykazac. [/latex]
