Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability (resit) / 4 / a / iii

Let $S_N=N^{-3}{\sum }_{n=1}^N{n}^2$. The expression in part (ii) givesBecause $3x^2$ is continuous on $[0,1]$, the composite trapezoidal rule converges to its definite integral, which is $1$ by part (i). In fact its standard composite trapezoidal error bound is $|T_N-1|\le 1/(2N^2)$, since $f^{\prime \prime }=6$. Therefore