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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
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