All i really want to know is does the series 1+(1/2)+(1/3)+(1/5)+ ... +(1/n). When n is a prime , converge. and if it does can anyone explain why.
Thanks for your help
Define to be the primes . For a given (sufficiently large) let be the primes .
Notice that .
Therefore, we see that .
By the fundamental theorem of arithmetic any number be be written as where .
It follows that,
Since the harmonic series diverges it means .
We have shown,
Remember that .
What this means is that if then would be bounded.
This is impossible, it is not bounded.
Thus, the sum of prime reciprocals must diverge.
a very clever proof of this result is due to Paul Erdos. see "Second Proof" in here. note that in the "Lower estimate" part of the proof the inclusion is actually an equality.
although this doesn't change anything in the proof, but still the wikipedia author should have been more careful!