Rare Gems

Calculus Level 4

On a distant planet, there are many types of gemstones, numbered according to rarity:

The second gemstone is twice as rare as the first gemstone,
the third gemstone is thrice as rare as the second,
the fourth gemstone is four times as rare as the third, and so on.
That is, the nth gemstone is n times as rare as the (n - 1)th.

There are an infinite number of these gemstones. What proportion of gems are the first type, rounded to the nearest thousandth?


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