The Infinite Cylinder

Consider a Cylinder infinitely high. There are infinitely many liquids placed in it. Each of these liquids is assigned a density $p_{n}$ , where $n$ corresponds to the position of the liquid from the bottom, starting at $1$ . These liquids have a recurrence relationship $p_{n} =\frac{p_{n-1}}{n-1}$ . Find the pressure exerted by the entire system at the bottom of the cylinder.

Note: The formula for liquid pressure is $hpg$ .