wykaż że ciąg określony podanym wzorem jest rosnący. Cn=[latex] frac{n-1}{2n+1} [/latex]

[latex]\c_{n+1}=frac{n}{2n+3} \ \c_{n+1}-c_n=frac{n}{2n+3}-frac{n-1}{2n+1}=frac{n(2n+1)-(n-1)(2n+3)}{(2n+1)(2n+3}= \ \frac{2n^2+n-2n^2-3n+2n+3}{(2n+1)(2n+3}=frac{3}{(2n+1)(2n+3}>0 implies{c_n} earrow[/latex]

[latex]c_{n+1}= frac{n+1-1}{2(n+1)+1}= frac{n}{2n+3}\c_{n+1}-c_{n}= frac{n}{2n+3}- frac{n-1}{2n+1}= frac{n(2n+1)-(n-1)(2n+3)}{(2n+3)(2n+1)}=\= frac{2n^2+n-(2n^2+3n-2n-3)}{4n^2+2n+6n+3}= frac{2n^2+n-2n^2-n+3}{4n^2+8n+3}= frac{3}{4n^2+8n+3}>0 [/latex]