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 Incoming links: Cosecant expansion near zero

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / 20 / b / Solution 2026-10-18

Let

u = x^{- 5}

, which tends to zero as

∣ x ∣ \to \infty

. Factor the denominator and use the cosecant expansion near zero:

g (x) = x^{2} \frac{u}{s i n u}, \frac{u}{s i n u} = (1 - \frac{u ^{2}}{6} + \frac{u ^{4}}{120} + O (u^{6}))^{- 1} = 1 + \frac{u ^{2}}{6} + \frac{7 u ^{4}}{360} + O (u^{6}) .

Substitution gives the first two nonzero terms and the requested omitted order:

g (x) = x^{2} + \frac{1}{6 x ^{8}} + O (x^{- 18}), ∣ x ∣ \to \infty.

More precisely the next term is

7/ (360 x^{18})

, followed by

O (x^{- 28})

. The outside factor

x^{2}

is essential when converting the remainder order from powers of

u

to powers of

x

.

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