Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Discrete mathematics, probability and continuous mathematics / 5 / b / iii / Solution

For $N$ equal intervals, the composite trapezoidal rule isAdding the missing $N^2/N^3$ term therefore gives the exact relationSince $x^2$ is continuous, its Riemann sums, and hence its endpoint-averaged trapezoidal sums, converge to its definite integral. Thus $T_N\to 1/3$ and $1/(2N)\to 0$, givingNo pre-existing formula for the sum of squares is needed.