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 Incoming links: Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 6 / a / iii

Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 6 / b / v / Solution 2026-10-25

By part iii,

X

is an integer with probability zero. Thus

⌈ X ⌉ = ⌊ X ⌋ + 1

almost surely, and

E [⌈ X ⌉] = ζ (t - 1) + 1.

The pointwise inequalities

⌊ X ⌋ \leq X \leq ⌈ X ⌉

imply

ζ (t - 1) \leq \frac{t - 1}{t - 2} \leq ζ (t - 1) + 1.

Putting

s = t - 1 > 1

and rearranging yields

\frac{1}{s - 1} \leq ζ (s) \leq \frac{s}{s - 1} .

This extracts bounds for an infinite series from simple rounding inequalities.

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