No Prime $E[{N}_{x}]$

Consider again,

${\prod _{ 1 }^{ { N }_{ x }}{ (\frac { 1 }{ 2 } )(\frac { 2 }{ 3 } )(\frac { 4 }{ 5 } )(\frac { 6 }{ 7 } )...\frac{{P}_{N}-1}{{P}_{N}}}}$

This term can be evaluated for all ${N}_{x}$ but the summation of its product with ${N}_{x}$ is not.  It is possible to evaluate $E[{N}_{x}]$ for a bounded set but not for all primes.  The proof that there are infinite numbers of primes suggests that the tail of the Probability ${N}_{x}$of  being Prime graph is never zero.  And as ${N}_{x}$ tends towards infinity, the term

$\frac{{N}_{x}-1}{{N}_{x}}=1-\frac{1}{{N}_{x}}$ is divergent with respect to $\int { (1-\frac { 1 }{ { N }_{ x } } ) } d{ N }_{ x }$.