Bug in infinite sum

*To*: mathgroup at smc.vnet.net*Subject*: [mg127388] Bug in infinite sum*From*: "Dr. Wolfgang Hintze" <weh at snafu.de>*Date*: Fri, 20 Jul 2012 23:42:20 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net

Consider this nice sum s[x_] = Sum[Binomial[x, k], {k, 0, Infinity}] 2^x and that, (with the odd terms only) u[x_] = Sum[Binomial[x, 2*k + 1], {k, 0, Infinity}] 2^(-1 + x) But now the even terms only ... and the surprise: g[x_] = Sum[Binomial[x, 2*k], {k, 0, Infinity}] (Sqrt[Pi]*Gamma[1 + x]*GegenbauerC[x, 1/2 - x, 1])/ (2^x*Gamma[1/2 + x]) looks complicated, but let's see FullSimplify[g[x], x > 0] 2^x*Cos[Pi*x] much simpler, but definitely wrong (giving e.g. 0 for x=1/2) Of course g should be s - u = 2^x -1/2 2^s = 1/2 2^s. Best regards, Wolfgang

**Follow-Ups**:**Re: Bug in infinite sum***From:*Bob Hanlon <hanlonr357@gmail.com>