 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-2/11h/f/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Write

f = Q_{2 n - 1} + E_{n}

, where

Q_{2 n - 1}

contains the terms through degree

2 n - 1

. On

[- 1, 1]

∣ E_{n} (x) ∣ \leq K \sum_{m = 2 n}^{\infty} R^{- m} = \frac{K R ^{- 2 n + 1}}{R - 1} .

The quadrature error of

Q_{2 n - 1}

is zero. Bounding the integral and quadrature errors of

E_{n}

separately, using positive weights with total

\int r

, gives exactly

\int f r - j \sum A_{j} f (ω_{j}) \leq \frac{2 K R ^{- 2 n + 1}}{R - 1} \int r .

 New to topics? Read the docs here!