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.
>
>
> 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

```

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