In the book "An Introduction to Number Theory" by Apostol there is a proof given by Clarkson, that $ \sum\_{n=1}^{\infty} {1 \over {p\_m}} $ diverges.

It is assumed $ \sum\_{m=k+1}^{\infty} {1 \over {p\_m}} < {1 \over 2} $ for a certain $ k \in \mathbb{N} $ . 
Then it defined $ Q $ as: $ Q = \prod\_{i=1}^{m} p\_m $ 

And reaches the equation: $ \sum\_{n=1}^{r} {1 \over 1+n\*Q} \le \sum\_{t=1}^{\infty} (\sum\_{m=k+1}^{\infty} {1 \over p\_m}) ^t $, which is perfectly fine. 

Now it is claimed that the "right side includes among its terms all the terms of the left." This I don't understand. 

Well, it is obvious for $ n = 1 $ that $ n\*Q+1 $ is a new prime and thus included on the right side. However I don't see how the quoted statement above holds, when $1+Q\*n$ is not prime. 

Though I find it clear, that all prime factors of $1+n\*Q$ appear in the sum of the right side.

I would be very happy, if someone could clear up this misunderstanding, as the problem seems rather basic and I still fail to get it right...