Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 18 / a / ii

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- Solution ii

Solution

Index the stated geometric progression by

x_{i} = (1 + ϵ)^{i}

,

i = 0, 1, \dots

. Then

Δ x_{i} = ϵ (1 + ϵ)^{i}

, and its left-endpoint contribution is

f (x_{i}) Δ x_{i} = (1 + ϵ)^{- (1 + α) i} ϵ (1 + ϵ)^{i} = ϵ [(1 + ϵ)^{- α}]^{i} .

Hence the approximation is a geometric series with

first term ϵ, q = (1 + ϵ)^{- α}, U_{ϵ} = \frac{ϵ}{1 - ( 1 + ϵ ) ^{- α}} .

Since

ϵ > 0

and

α > 0

, the ratio satisfies

0 < q < 1

and the series converges. The integrand is decreasing, so the left-endpoint sum is an upper bound for the improper integral.

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