Testing Mathjax

It's well known that the sum of the reciprocals of the positive integers is infinite. It's less well known (ran across it a couple years ago) that the sum of the reciprocals of the primes is also infinite (Euler, 1737).

The linked article has proofs about the series $1/p_i$, but seems to only say that the divergence of the sum $\sum_{i=1}^{n} 1/p_i$ is greater than $\log\log(n+1)$.

I'd bet that $$\lim_{n\to \infty} {{\sum_{i=1}^{n} 1/p_i} \over {\sum_{i=1}^{n} 1/i} } \to 0$$ it seems obvious -- but I'm not sure how to prove that yet. Euler could probably have done it in his sleep. I'd need to mull over their approaches for a while.

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It looks like this works

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UPDATE: Yep, it's pretty obvious. The difference between the prime sum and $\log(\log(n))$ is finite, and so the numerator is close to $\log(\log$ and the denominator to $\log$ so the ratio tends to 0. Anyhow, this Mathjax tool seems to pass the initial tests.