The inverse sum of all prime numbers diverges

A great fact that Euler proved about the ***Harmonic Series*** is that:

Which at the core comes from the fact of that for some a>1:

Which then will let us rewrite the left side of the quation that Euler proved as:

Which then on it’s expansion will generate indeed all numbers because with prime numbers we can form any number and every number as only one way to be written as a product of prime numbers.

But what happens if we wrote this product only partially? Well, it would form many numbers, but the numbers it would form would include many more numbers than there will be included in the partial sum of the Harmonic Series up to the last prime number in the product, in other words:

But, if we were to generalize this up to some prime number say pk, where pk is the kth prime number, then we would have:

And as you would as well notice the sum partial sum of the Harmonic Series is larger than the integral of 1/x from 1 all the way up to pk which would be just the ln(pk) and therefore one has that:

And if we took the natural logarithm on both sides we would then have:

And noticing now for all of these logarithms they will be strictly larger than -2/p where p is a prime number, then:

Which is twice the sum of the inverse of prime numbers all the way up to the kth prime number, and then we can say that:

And as we let it go to infinity the right side of the inequality goes as well to infinity and because the left is larger than the right, the left must also go to infinity and therefore we have proven that the sum of all inverse prime numbers diverges.

Soli Deo Gloria.