Question

Evaluate the limit by first recognizing the sum as the Riemann sum for a function defined on the interval [0,1]: lim as n approaches infinity (1/n)((sqrt(1/n)+(sqrt(2/n)+(sqrt(3/n)+...............+(sqrt(n/n))

Answer 1

$\lim_{n \to +\infty} \left ( \dfrac{1}{n} \cdot \left ( \sqrt{ \dfrac{1}{n} } + \sqrt{ \dfrac{2}{n} } + \sqrt{ \dfrac{3}{n} } + \cdots + \sqrt{ \dfrac{n}{n} }} \right ) \right ) =$

$\dfrac{1}{n} \sum_{i = 1}^n \left ( \sqrt{ \dfrac{i}{n} } \right ) = \int\limits^1_0 { \sqrt{x} } \, dx = \left [ \frac{2}{3}x^{ \frac{3}{2} } \right ]^{1}_{0} = \dfrac{2}{3} \cdot 1^{ \frac{3}{2} } - \dfrac{2}{3} \cdot 0^{ \frac{3}{2} } = \boxed{ \dfrac{2}{3} }$