We've shown elsewhere that there are an infinite number of primes. To show that with p prime, the series
∑p1/p = 1/2 + 1/3 + 1/5 + ... + 1/p + ...
diverges.
Outline of Proof: We first establish that the harmonic series
H = ∑n1/n = 1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ...
with n a natural number, diverges. We observe that for x > 1, ∫1x 1/t dt = ln x, which diverges.
Using a form of the Sieve of Eratosthenes, we subtract from H the series ∑n1/2n = 1/2 + 1/4 + 1/6 + ... + 1/2n + ... = H/2.
The difference is H/2 = 1 + 1/3 + 1/5 + 1/7 + 1/9 + ...
Next we subtract the series ∑n1/3n = 1/3 + 1/6 + 1/9 + ... + 1/3n + ... = H/3.
The difference is H/6 = 1 + 1/5 + 1/7 + 1/11 + ...
Note: At each stage in the subtractions, the difference, which is always a fraction of H, still diverges.
We can show by induction that if the sum still diverges after the ith such subtraction, then it will also diverge after the i+1st subtraction, and the result is established.
QED