Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability / 6 / a / ii / Solution

Write the sum in the question as the right Riemann sumRelative to the composite trapezoidal rule, the only change is in the two endpoint weights:The integrand $f(x)=1/(1+x^2)$ is continuous on $[0,1]$, so $T_N\to I$. Since an antiderivative is $\arctan x$,Thus