For $a\in R$ . calculate $\lim\limits_{n\to\infty}\frac{1}{n}$( $( a+\frac{1}{n})^2 + (a +\frac{2}{n})^2+.....+( a + \frac{n-1}{n})^2$)

Question

For $a\in R$ . calculate

$\lim\limits_{n\to\infty}\frac{1}{n}( $$( a+\frac{1}{n})^2 + (a +\frac{2}{n})^2+.....+( a + \frac{n-1}{n})^2$)

As i was thinking about the formula $1^2 + 2^2 + 3^2 +....+n^2= \frac {n(n+1)(2n+1) }{6}$

But here i don't know how to solved this tough problem as also i don't know to approach this problem..as i was thinking a lots but i disn't get any clue and any hints

Thanks In advance

Answer 1

Another method would be to convert the summation to an integral: $$\sum_{k=1}^{n-1}\frac{1}{n}\left(a+\frac{k}{n}\right)^2=\int_{a}^{a+1}x^2 dx$$ $$=\frac{\left( a+1 \right)^3- a^3 }{3}= a^2 +a +\frac{1}{3}$$

Answer 2

Hint:

$$\sum_{k=1}^{n-1}\left(a+\frac{k}{n}\right)^2=(n-1)a^2+\frac{2a}{n}\sum_{k=1}^{n-1}k+\frac{1}{n^2}\sum_{k=1}^{n-1}k^2$$