Let $q$ be a positive integer. For every positive integer $k$, define a number $c_{k}$ by the formula

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

Prove by induction that

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

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

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