[tex]\it n^2 \ \textless \ n(n+1) \ \textless \ (n+1)^2 \Rightarrow \sqrt{n^2} \ \textless \ \sqrt{n(n+1)} \ \textless \ \sqrt{(n+1)^2} \Rightarrow
\\\;\\
\Rightarrow n \ \textless \ \sqrt{n(n+1)} \ \textless \ n+1 \Rightarrow [\sqrt{n(n+1)}] = n[/tex]
[tex]\it \{\sqrt{n(n+1)}\} = \sqrt{n(n+1)} -n = \dfrac{(\sqrt{n(n+1)} -n)(\sqrt{n(n+1)} +n)}{\sqrt{n(n+1)} +n}
\\\;\\ \\\;\\
= \dfrac{n^2+n-n^2}{\sqrt{n^2+n} +n} =\dfrac{n}{n\left(\sqrt{1+\dfrac{1}{n}}+1 \right)} =\dfrac{1}{\sqrt{1+\dfrac{1}{n}}+1} \longrightarrow \dfrac{1}{2} =0,5[/tex]
