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 $p_1 = 2$,

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

No, it does not converge. See this: Proof of divergence of sum of reciprocals of primes.

In fact it is known that $$\sum_{p \le x} \frac{1}{p} = \log \log x + A + \mathcal{O}(\frac{1}{\log^2 x})$$

Related: Proving $\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} \geq \frac{1}{2}\log{x} -\log{\log{x}}$

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!