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