Science:Math Exam Resources/Courses/MATH103/April 2014/Question 04 (a)/Solution 1

By definition of the integral (see Hint 3),

$\lim _{n\to \infty }{\frac {1}{n}}\sum _{k=1}^{n}f\left({\frac {k}{n}}\right)=\int _{0}^{1}f(x)\mathrm {d} x.$

Comparing to the original sum $\sum _{k=1}^{n}\left({\frac {k}{n}}\right)^{2}{\frac {1}{n}}$ we have that $f(x)=x^{2}$. Therefore,

$\lim _{n\to \infty }{\frac {1}{n}}\sum _{k=1}^{n}\left({\frac {k}{n}}\right)^{2}=\int _{0}^{1}x^{2}\mathrm {d} x=\left.{\frac {1}{3}}x^{3}\right|_{0}^{1}={\frac {1}{3}}.$