Re: Sum of 1/Primes

*To*: mathgroup at smc.vnet.net*Subject*: [mg36946] Re: [mg36910] Sum of 1/Primes*From*: Andrzej Kozlowski <andrzej at platon.c.u-tokyo.ac.jp>*Date*: Thu, 3 Oct 2002 00:16:21 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

The fact that Sum[1/Prime[n], {n, 1,Infinity}]==Infinity is a rather famous theorem of Euler. It implies that there must be infinitely many primes (otherwise the sum would be finite), and was the beginning of a vast area of mathematics, which includes such concepts as Dirichlet series, Riemann's zeta function etc. Andrzej Kozlowski Yokohama, Japan http://www.mimuw.edu.pl/~akoz/ http://platon.c.u-tokyo.ac.jp/andrzej/ On Wednesday, October 2, 2002, at 04:31 PM, Matthias.Bode at oppenheim.de wrote: > Dear Colleagues, > > I calculated: > > Sum[1/Prime[n], {n, 15000}] // N > > Result: 2.74716 > > Now I wonder if this sum will converge or keep on growing, albeit very > slowly. > > Best regards, > > Matthias Bode > Sal. Oppenheim jr. & Cie. KGaA > Koenigsberger Strasse 29 > D-60487 Frankfurt am Main > GERMANY > Tel.: +49(0)69 71 34 53 80 > Mobile: +49(0)172 6 74 95 77 > Fax: +49(0)69 71 34 95 380 > E-mail: matthias.bode at oppenheim.de > Internet: http://www.oppenheim.de > > > >