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Re: Re: Disturbing products

  • To: mathgroup at
  • Subject: [mg33366] Re: [mg33329] Re: Disturbing products
  • From: "Fred Simons" <f.h.simons at>
  • Date: Sun, 17 Mar 2002 05:33:25 -0500 (EST)
  • References: <>
  • Sender: owner-wri-mathgroup at

----- Original Message -----
From: Tomas Garza <tgarza01 at>
To: mathgroup at
Subject: [mg33366] [mg33329] Re: Disturbing products

> In a recent message to this group ([mg33251] Animation), Juan Erfa posed
> a question which has not received any comment from this group.
> Essentially, the problem is
> In[1]:=
> Product[(1 + 1/n)^2, {n, Infinity}]
> Out[1]=
> 1
> In[2]:=
> Limit[Product[(1 + 1/n)^2, {n, k}], k -> Infinity]
> Out[2]=
> Infinity
> The first output is wrong, while the second is right.
> Any idea about this apparent bug?
> Tomas Garza
> Mexico City

It seems to be a consequence of a more general bug:

Product[1 + a/n + b/n^2, {n, 1, Infinity}]

1/(Gamma[(2 + a - Sqrt[a^2 - 4*b])/2]*
  Gamma[(2 + a + Sqrt[a^2 - 4*b])/2])

This is incorrect for a!=0. For a=0 the result is correct, though the form
is different from that obtained by a direct computation:

% /. a->0

1/(Gamma[(2 - 2*Sqrt[-b])/2]*
  Gamma[(2 + 2*Sqrt[-b])/2])

Product[1+b /n^2, {n, 1, Infinity}]




When we replace in In[3] n^2 with n^3 of n^4, we get incorrect results as

There is another, maybe related, bug in Product:

Product[b+ a /n, {n, 1, Infinity}]

Gamma[(a + b)/b]/E

This is incorrect for all real values of a and b.

Finally something very astonishing (Mathematica 4.1 onder Windows 98):

Sum[n, {n, 1, \[Infinity]}]


Fred Simons
Eindhoven University of Technology

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