I know the $\sum_{n=1}^\infty \frac{1}{\text{Prime[$n$]}}$ does not converge, but what about the  following series? $$\sum_{n=1}^\infty\frac{1}{\text{Prime[Prime[$n$]]}}$$  (Where $\text{Prime[$n$]}$ is the $n$th prime number.)  I ran some calculation on Mathematica, but I am not sure in the answer gets very close to 1.  Can anyone help with that?  NB: I an not a professional mathematician. I am just an amateur.
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 ?
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