Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 11H / f

Table of contents
- Solution f

Solution

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 .

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