What is the value of $\sum\limits\_{i=1}^\infty\dfrac{1}{p\_{p\_i}}$ where $p\_n$ is the nth prime (and so $p\_{p\_n}$ is the $k$th prime, where $k$ is the $n$th prime) ?

Thus $\frac{1}{3}+\frac{1}{5}+\frac{1}{11}+...=$ ??

How do I efficiently decide if $\sum\_{i=1}^\infty\dfrac{1}{p\_{p\_i}}\gt1$ is true without using a computer ?

I assume this infinite sum is an irrational number. Has this already been proven ?