TITLE: Is the series $X =\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+..$ convergent or divergent.
QUESTION [6 upvotes]: The series is the reciprocal of twin primes. Let $Y=(y_n)$be the series of reciprocal of natural numbers.  Now if I use the comparison test we can see that each term of $0 <(x_n) < (y_n)$ .So the divergence of series of reciprocal of  natural number should also imply the same for $Y$. Now the problems of my proof:
$1)$ I probably have used the comparison test in a wrong sense
$2)$ I don't know how many twin primes are there 
Can someone help me to understand the intuition behind the statement. If it converges or diverges why? I don't need the entire proof but some hint would do..

REPLY [4 votes]: The result and its proof have already been provided in the comments. To answer your question about the intuition behind it: Based on the random model of the primes, the first Hardy-Littlewood conjecture estimates the density of twin primes (and similar prime constellations) at $x$ to be of the order of $\frac1{\ln^2x}$. Based on this estimate and the fact that the integral
$$
\int_a^\infty \frac1x\frac1{\ln^2x}\mathrm dx=\frac1{\ln a}
$$
converges, we should expect the sum of the reciprocals of the twin primes to converge.