_n (3)/(n)4+(2+(1)/(n))^2+(2+(2)/(n))^2++(3-(1)/(n))^2 is equal to :

Question

$\lim\limits_{n \rightarrow \infty} \frac{3}{n}\left\{4+\left(2+\frac{1}{n}\right)^2+\left(2+\frac{2}{n}\right)^2+\ldots+\left(3-\frac{1}{n}\right)^2\right\}$ is equal to :

Accepted Answer

$$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 0}^{n - 1} {{3 \over n}{{\left( {2 + {r \over n}} \right)}^2}} $$

$$ = \int_0^1 {3{{(2 + x)}^2}\,dx} $$

$$ = \left. {3\,.\,{{{{(2 + x)}^3}} \over 3}} \right|_0^1$$

$= {3^3} - {2^3} = 19$

Solutions · 1