slightly OT: harmonic series revisited...

[Rob Arthan]

On Tuesday 13 Dec 2005 12:26 pm, hv at crypt.org wrote:
> "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.
>...
>
> 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.
>

Nice argument! You only need the second bit because Euler proved that P 
diverges. There is some discussion and references at 
http://mathworld.wolfram.com/PrimeSums.html.

Regards,

Rob.