# Irrationality of $(p_1 + p_2)/(p_1 \times p_2) + (p_1 + p_2 + p_3)/(p_1 \times p_2 \times p_3) + \ldots$, where $p_i$ is an $i$-th prime

Assuming that $$p_i$$ denotes an $$i$$-th prime, the number $$x$$ is defined as follows:

$$\begin{array}{l} x = \frac{{{p_1} + {p_2}}}{{{p_1} \times {p_2}}} + \frac{{{p_1} + {p_2} + {p_3}}}{{{p_1} \times {p_2} \times {p_3}}} + \frac{{{p_1} + {p_2} + {p_3} + {p_4}}}{{{p_1} \times {p_2} \times {p_3} \times {p_4}}} + \ldots = \\ = \frac{{2 + 3}}{{2 \times 3}} + \frac{{2 + 3 + 5}}{{2 \times 3 \times 5}} + \frac{{2 + 3 + 5 + 7}}{{2 \times 3 \times 5 \times 7}} + \ldots = \\ = 1.2612275803899663618 \ldots \end{array}$$

Question: is it possible to prove that $$x$$ is irrational? If yes, how?