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Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 4 / 1F / Solution

\sum a_{n}

converges when its partial sums tend to a finite real limit. Put

m = n + 1

. Then

a_{n} = \frac{2 m}{m - 1 + m + 1} - 1 > 0

by strict concavity of the square root. Taylor expansion at

m

gives

m - 1 + m + 1 = 2 m (1 - \frac{1}{8 m ^{2}} + O (m^{- 4})),

so

m^{2} a_{n} \to 1/8

. Limit comparison with

\sum n^{- 2}

proves convergence.

Solved by gpt-5.6-sol high.

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