slightly OT: harmonic series revisited...

[hv at crypt.org]

"santi_spadaro at virgilio.it" <santi_spadaro at virgilio.it> wrote:
:Anybody knows an answer (and a neat way to show that the answer is 
:true)?
:
:"Define a_n = 1/n if n is composite and a_n = -(1/n) if n is 
:prime. Does the series of a_n (sum from n to infinity of a_n) diverges?"

I'm not sure offhand whether P = sum{1/p} diverges, but it doesn't matter.

We know that N = sum{1/n} diverges, so if P converges A = sum{a_n} clearly
must diverge - we are subtracting (twice) a finite sum from an infinite one.

If P diverges, consider the set {p, 2p, 3p, 4p, 6p}; this avoids collisions
for all odd primes p, and the contribution to A for these 5 numbers is
(-1 + 1/2 + 1/3 + 1/4 + 1/6)/p = 1/4p, so A > P/4, and so A again diverges.

Hugo