Does the sum of reciprocals of primes converge?

Is this series known to converge, and if so, what does it converge to (if known)?

Where $p_n$ is prime number n, and $p_1 = 2$,

$$\sum\limits_{n=1}^{\infty} \frac{1}{p_n}$$

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A cultural note: Euler was (as far as I know) the first person to observe that this series diverges (and without assuming a priori that it was an infinite series), thus obtaining a new proof of the infinitude of the primes. This was a precursor to Dirichlet's work in which he proved the infinitude of primes in arithmetic progressions, and then Riemann, Hadamard, and de la Vallee Poussin's work leading to the prime number theorem. –  Matt E Dec 31 '10 at 4:28
Unfortunately, the same technique does not apply for other problems. For instance, the sum of reciprocals of the twin primes converges and the existence of infinitely many twin primes remains open. –  lhf Dec 31 '10 at 6:57

In fact it is known that $$\sum_{p \le x} \frac{1}{p} = \log \log x + A + \mathcal{O}(\frac{1}{\log^2 x})$$
I would like to note that this implies that according the Müntz-Szász Theorem that every continuous function in $[0,1]$ is a uniform limit of polynomials whose exponents are prime numbers!