∫ 0 2 f ( x ) d x = 2 , 67 {\displaystyle \int \limits _{0}^{2}f(x)dx=2,67}
8 3 {\displaystyle {\dfrac {8}{3}}}
= 4 3 ⋅ n + 1 n ⋅ 2 n + 1 n {\displaystyle ={\dfrac {4}{3}}\cdot {\dfrac {n+1}{n}}\cdot {\dfrac {2n+1}{n}}}
= 4 3 ⋅ ( 1 + 1 n ) ( 2 + 1 n ) {\displaystyle ={\dfrac {4}{3}}\cdot {\begin{pmatrix}1+{\dfrac {1}{n}}\end{pmatrix}}{\begin{pmatrix}2+{\dfrac {1}{n}}\end{pmatrix}}}
O n = 2 n [ ( 2 n ) 2 + ( 2 ⋅ 2 n ) 2 + ( 3 ⋅ 2 n ) 2 + . . . + ( n ⋅ 2 n ) 2 ] {\displaystyle O_{n}={\frac {2}{n}}{\begin{bmatrix}{\begin{pmatrix}{\dfrac {2}{n}}\end{pmatrix}}^{2}+{\begin{pmatrix}2\cdot {\dfrac {2}{n}}\end{pmatrix}}^{2}+{\begin{pmatrix}3\cdot {\dfrac {2}{n}}\end{pmatrix}}^{2}+...+{\begin{pmatrix}n\cdot {\dfrac {2}{n}}\end{pmatrix}}^{2}\end{bmatrix}}}
= 2 3 n 3 [ 1 2 + 2 2 + 3 2 + . . . + n 2 ] {\displaystyle ={\dfrac {2^{3}}{n^{3}}}{\begin{bmatrix}1^{2}+2^{2}+3^{2}+...+n^{2}\end{bmatrix}}}
1 2 + 2 2 + 3 2 + . . . + z 2 = 1 6 z ( z + 1 ) ( 2 z + 1 ) {\displaystyle 1^{2}+2^{2}+3^{2}+...+z^{2}={\dfrac {1}{6}}z{\begin{pmatrix}z+1\end{pmatrix}}{\begin{pmatrix}2z+1\end{pmatrix}}}