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I was trying to find a series that converged to $1$ and represented the amount of composites up to infinity as a fraction. I had originally tried to sum an infinite series of fractions while guarding for overlap, but found that I was overcompensating for overlap in, Sum of an infinite series of fractions. One answer to that question involved the following: the series described above can be written like this

$\displaystyle\;1 - \prod_{k=1}^n\left(1 - \frac{1}{p_k}\right)\;$,

where $p_k$ represents the $k^{th}$ prime.

I'm wondering whether this is correct, and why? Note that this is not a duplicate of Sum of an infinite series of fractions because I am asking for a full explanation as to why the product represents what it does.

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    By the prime number theorem there are [$\pi(n)$](https://en.wikipedia.org/wiki/Prime-counting_function) $\sim \frac{n}{\log(n)}$ primes below $n$ so the fraction of composite numbers is $1 - \frac{\pi(n)}{n} \sim 1 - \frac{1}{\log(n)}$. – 2017-01-14
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    @Kibble Thanks, would you mind expanding this into a full answer? Why is the other product wrong? – 2017-01-14
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    $1 - \prod\limits_{k = 1}^n \bigl(1 - \frac{1}{p_k}\bigr)$ is the fraction of natural numbers that are divisible by (at least) one of the first $n$ primes. – 2017-01-14
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    Possible duplicate of [Sum of an infinite series of fractions](http://math.stackexchange.com/questions/2044965/sum-of-an-infinite-series-of-fractions) – 2017-01-15

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