It is well-known that not only does the arithmetic progression $\{ak+b\}_{k \in \mathbb{Z}^{+}}$ contain infinitely many prime numbers, but also that the series of the reciprocals of those primes diverges. The answer to the OP's question can be obtained now from the following general result:

*If* $\{a_{i}\}_{i \in \mathbb{N}}$ *is a strictly increasing sequence of natural numbers such that the series* $\sum_{i=1}^{\infty} \frac{1}{a_{i}}$ *diverges, then the unending decimal fraction $\alpha$ formed by juxtaposing the successive terms of the sequence* $\{a_{i}\}_{i \in \mathbb{N}}$ *represents an irrational number.*

For a proof of this theorem see D. J. Newman, R. Breusch, and F. Herzog. Solution to problem 4494. *Amer. Math. Monthly* 9 (**60**), Nov. 1953, pp. 632-633. or N. Hegyvari. On some irrational decimal fractions. *Amer. Math. Monthly* 8 (**100**), Oct. 1993, pp. 779-780.