Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 2G / c

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- Solution c

Solution

Let

x = \sum_{j = 1}^{\infty} \frac{θ _{j}}{j !}

and suppose

x = p / q

. Choose

n \geq max (q, 2)

such that some

θ_{j} = 1

for

j > n

. Then

n! x

is an integer, while its first

n

terms also sum to an integer. Their difference is strictly positive and satisfies

0 < n! \sum_{j > n} \frac{θ _{j}}{j !} \leq \sum_{r = 1}^{\infty} \frac{1}{( n + 1 ) ^{r}} < 1.

This cannot be the difference of two integers, so

x

is irrational.

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