SOME PROPERTIES OF THE SEQUENCE OF PRIME NUMBERS

Let $p_n$ be the $n$-th prime number and $x_n=p_{n+1}^{;n+1}/p_n^{;n}$. We show that the sequence $(x_n)_{ngeN}$ is not monotonic for any integer $N>1$ and that the series $sumlimits_{n=1}^{+infty} 1/x_n$ is divergent. Related series are studied as well.