Fastest way to calculate inverse consecutive numbers?

Question

I am trying to find my way through a problem involving inverse numbers, for a game.

I need to reduce the number of calculations of the following product:

$\frac{1}{2}\times\frac{1}{3}\times \dots \times\frac{1}{n}$

with n being variable. I wondered if there were a fastest way than calculating every inverse.

Question:

is there a known formula to help?

Note: I am not asking for a programming algorithm, just a theoretical one, an idea.

In advance, thank you,

Edit:

the result must be precise to the closest natural.