The approximation can be justified by applying the procedure of the integral test for convergence to get .
A similar procedure will give you , , , and so on.
Edit. See corrections below.
Is there a meaningful infinite set , such that
diverges ~
Like maybe let ~prime,~p\equiv1\mod4\}" alt="A=\{p\in\mathbb{N}~prime,~p\equiv1\mod4\}" />
But, for all I know, that could converge. (I know that if ~prime,~p+2~also~prime\}" alt="A=\{p~prime,~p+2~also~prime\}" />, then the series does converge.)
Actually, now that I think about it, that sum must converge, because
EDIT: Although, the above calculation seems to imply that if we let ~prime,~p\equiv3\mod4\}" alt="B=\{p~prime,~p\equiv3\mod4\}" /> then diverges.