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Bug in analytical sum

• To: mathgroup at smc.vnet.net
• Subject: [mg100244] Bug in analytical sum
• From: Sebastian Meznaric <meznaric at gmail.com>
• Date: Fri, 29 May 2009 20:57:06 -0400 (EDT)

Consider the following sum

f[k_] := Sum[n! (-I \[Alpha])^n Cos[\[Phi]/2]^n \[Alpha]^n Sin[\[Phi]/
2]^n, {n,
    0, k}]/Sum[n! \[Alpha]^(2 n) Cos[\[Phi]/2]^(2 n), {n, 0, k}]

I am interested in taking alpha and k to infinity. Now clearly for
finite k this is just a rational function in alpha. So if we want to
take alpha to infinity we should get

(-i Tan[\[Phi]/2])^k.

But try this in Mathematica Limit[f[k], \[Alpha] -> \[Infinity]] and
you will get I Cot[\[Phi]/2]. This is Mathematica 7.0.0.

The reason seems to be that Mathematica evaluates the sum first and
obtains a fraction consisting of the incomplete gamma functions and Ei
integrals. It seems that the limits of those functions are not taken
properly.

• Follow-Ups:
  • Re: Bug in analytical sum
    • From: danl@wolfram.com