Paper 4, Section II, 7E7 \mathrm{E}

Numbers and Sets | Part IA, 2021

(a) Prove that every real number α(0,1]\alpha \in(0,1] can be written in the form α=\alpha= n=12bn\sum_{n=1}^{\infty} 2^{-b_{n}} where (bn)\left(b_{n}\right) is a strictly increasing sequence of positive integers.

Are such expressions unique?

(b) Let θR\theta \in \mathbb{R} be a root of f(x)=αdxd++α1x+α0f(x)=\alpha_{d} x^{d}+\cdots+\alpha_{1} x+\alpha_{0}, where α0,,αdZ\alpha_{0}, \ldots, \alpha_{d} \in \mathbb{Z}. Suppose that ff has no rational roots, except possibly θ\theta.

(i) Show that if s,tRs, t \in \mathbb{R} then

f(s)f(t)A(max{s,t,1})d1st.|f(s)-f(t)| \leqslant A(\max \{|s|,|t|, 1\})^{d-1}|s-t| .

where AA is a constant depending only on ff.

(ii) Deduce that if p,qZp, q \in \mathbb{Z} with q>0q>0 and 0<θpq<10<\left|\theta-\frac{p}{q}\right|<1 then

θpq1A(1θ+1)d11qd.\left|\theta-\frac{p}{q}\right| \geqslant \frac{1}{A}\left(\frac{1}{|\theta|+1}\right)^{d-1} \frac{1}{q^{d}} .

(c) Prove that α=n=12n!\alpha=\sum_{n=1}^{\infty} 2^{-n !} is transcendental.

(d) Let β\beta and γ\gamma be transcendental numbers. What of the following statements are always true and which can be false? Briefly justify your answers.

(i) βγ\beta \gamma is transcendental.

(ii) βn\beta^{n} is transcendental for every nNn \in \mathbb{N}.

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