I was recently trying to create my own unique proof that the harmonic series diverges, so what I did was realize that within the harmonic series is the reciprocal of powers of primes and for every prime, every reciprocal of the powers of that prime also are contained. By regrouping, you can show that for every prime, the powers of that prime add up to 1/(p-1) so we know that for every prime, the harmonic series at least
contains 1/(p-1) which means that the harmonic series is greater than the sum from 1 to infinity of 1/(p-1) where the sum is over the primes. How would I go about showing that this sum diverges?
Sorry about formatting I am currently doing this on a mobile device.
No comments:
Post a Comment