Numbers and Sets | Part IA, 2008

Let qq be a positive integer. For every positive integer kk, define a number ckc_{k} by the formula

ck=(q+k1)q!(q+k)!c_{k}=(q+k-1) \frac{q !}{(q+k) !}

Prove by induction that

k=1nck=1q!(q+n)!\sum_{k=1}^{n} c_{k}=1-\frac{q !}{(q+n) !}

for every n1n \geqslant 1, and hence evaluate the infinite sumk=1ck\operatorname{sum} \sum_{k=1}^{\infty} c_{k}.

Let a1,a2,a3,a_{1}, a_{2}, a_{3}, \ldots be a sequence of integers satisfying the inequality 0an<n0 \leqslant a_{n}<n for every nn. Prove that the series n=1an/n\sum_{n=1}^{\infty} a_{n} / n ! is convergent. Prove also that its limit is irrational if and only if ann2a_{n} \leqslant n-2 for infinitely many nn and am>0a_{m}>0 for infinitely many mm.

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