Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 2 / 10 / b / iii

Use the stipulated interpolation $k_3\approx (k_2+k_4)/2$. The unrounded values give $k_3\approx 0.4728010040$, and henceThus a sensible reported estimate isUsing the rounded shifts from part (ii) instead gives $k_3\approx 0.473$ and a sum about $1.202059745$, with the same six-significant-figure estimate. The interpolation of shifts is an assumption of this approximation; the integral test bounds alone do not prove that many digits of accuracy.