Re: Re: Disturbing products
- To: mathgroup at smc.vnet.net
- Subject: [mg33366] Re: [mg33329] Re: Disturbing products
- From: "Fred Simons" <f.h.simons at tue.nl>
- Date: Sun, 17 Mar 2002 05:33:25 -0500 (EST)
- References: <200203160640.BAA18383@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
----- Original Message -----
From: Tomas Garza <tgarza01 at prodigy.net.mx>
To: mathgroup at smc.vnet.net
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:
In[3]:=
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:
In[4]:=
% /. a->0
1/(Gamma[(2 - 2*Sqrt[-b])/2]*
Gamma[(2 + 2*Sqrt[-b])/2])
In[5]:=
Product[1+b /n^2, {n, 1, Infinity}]
Sinh[Sqrt[b]*Pi]/(Sqrt[b]*Pi)
In[6]:=
FullSimplify[%-%%]
0
When we replace in In[3] n^2 with n^3 of n^4, we get incorrect results as
well.
There is another, maybe related, bug in Product:
In[7]:=
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):
In[8]:=
Sum[n, {n, 1, \[Infinity]}]
-1/12
Fred Simons
Eindhoven University of Technology
- References:
- Re: Disturbing products
- From: Tomas Garza <tgarza01@prodigy.net.mx>
- Re: Disturbing products