Sum of reciprocals of prime-index-primes Let $p_1=2$, $p_2=3$, $p_3=5$, $\ldots$ be an enumeration the prime numbers. If $q$ is a prime number, we call $p_q$ a prime-index-prime. A list of prime-index-primes can be found here.
My question is:

Does $$\sum_{i=1}^{\infty}\frac{1}{p_{p_i}}=\frac{1}{3}+\frac{1}{5}+\frac{1}{11}+\frac{1}{17}+\frac{1}{31}+\ldots$$ converges or diverges?

The graph below shows the values of the partial sums $S_n$ of the previous series from $n=1$ to $n=1184$, $n$ is the $x$-coordinate and $S_n$ is the $y$-coordinate.

We can't conclude much by the graph. But my guess is that it diverges because, perhaps, it grows very, very, slow.

Maybe it's interesting to see a proof of the convergence. From PNT we have that $$p_{n}\sim n\log\left(n\right)
 $$ as $n\rightarrow\infty
 $ so $$p_{p_{n}}\sim p_{n}\log\left(p_{n}\right)\sim n\log^{2}\left(n\right)
 $$ and we can observe that, using the integral test, that $$\sum_{n=2}^{\infty}\frac{1}{n\log^{2}\left(n\right)}\leq\frac{1}{2\log^{2}\left(2\right)}+\int_{2}^{\infty}\frac{dx}{x\log^{2}\left(x\right)}=\frac{1+2\log\left(2\right)}{2\log^{2}\left(2\right)}.$$
