Re: Re: infinite product

*To*: mathgroup at smc.vnet.net*Subject*: [mg65327] Re: [mg65309] Re: infinite product*From*: Devendra Kapadia <dkapadia at wolfram.com>*Date*: Mon, 27 Mar 2006 06:55:52 -0500 (EST)*References*: <6334718.1143190534402.JavaMail.jakarta@nitrogen.mathforum.org> <240320060230425202%bruck@math.usc.edu> <20060324081024.411$gC@newsreader.com> <200603251017.FAA02277@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

On Sat, 25 Mar 2006, Ronald Bruck wrote: > In article <20060324081024.411$gC at newsreader.com>, David W. Cantrell > <DWCantrell at sigmaxi.org> wrote: > >> Ronald Bruck <bruck at math.usc.edu> wrote: >>> In article >>> <6334718.1143190534402.JavaMail.jakarta at nitrogen.mathforum.org>, eugene >>> <jane1806 at mail.ru> wrote: >>> >>>> Could you please help me to calculate the following product >>>> \prod_{n=2}^{\infty} (n^2-1)/(n^2+1). >>>> >>>> In the case \prod (n^3-1)/(n^3+1) we can easily cancel out the >>>> multipliers in the numerator and denomonator and it can easily be >>>> proved that the values in this case is 2/3. But i have no ideas to deal >>>> with our case with squares. >>> >>> I wouldn't expect this to be anything simple, but I plugged it into >>> Mathematica anyway. The result may be very interesting to fans of >>> Mathematica: >>> >>> {(E^(LogGamma[2 - I] + LogGamma[2 + I])*Pi*Csch[Pi]* Gamma[1 + >>> System`SeriesDump`k]^2)/ >>> (20*Internal`ErdelyiBernoulliB[System`SeriesDump`k, 3 - I, 2 - >>> I]*Internal`ErdelyiBernoulliB[ System`SeriesDump`k, 3 + I, 2 + I]* >>> Gamma[(-2 - I) + System`SeriesDump`k]* Gamma[(-2 + I) + >>> System`SeriesDump`k]), (E^((-2*I)*System`SeriesDump`k*Pi + LogGamma[2 >>> - I] + LogGamma[2 + I])*Gamma[-2 - I]*Gamma[-2 + I]* Gamma[1 + >>> System`SeriesDump`k]^2)/ >>> (2*Internal`ErdelyiBernoulliB[System`SeriesDump`k, 3 - I, 2 - >>> I]*Internal`ErdelyiBernoulliB[ System`SeriesDump`k, 3 + I, 2 + I]* >>> Gamma[(-2 - I) + System`SeriesDump`k]* Gamma[(-2 + I) + >>> System`SeriesDump`k]), (E^(LogGamma[2 - I] + LogGamma[2 + >>> I])*Pi*Csch[Pi]* Gamma[1 + System`SeriesDump`k]^2)/ >>> (20*Internal`ErdelyiBernoulliB[System`SeriesDump`k, 3 - I, 2 - >>> I]*Internal`ErdelyiBernoulliB[ System`SeriesDump`k, 3 + I, 2 + I]* >>> Gamma[(-2 - I) + System`SeriesDump`k]* Gamma[(-2 + I) + >>> System`SeriesDump`k])} >>> >>> Question is, what are "System`SeriesDump`k" and "ErdelyiBernoulliB"? >>> The former is clearly some sort of abort, and the latter an internal >>> routine. >>> >>> The numerical value is approximately 0.272029. >> >> I've gotten garbage like that from Mathematica before too. But I don't in >> this case, using version 5.1: >> >> In[1]:= FullSimplify[Product[(n^2 - 1)/(n^2 + 1), {n, 2, Infinity}]] >> >> Out[1]= Pi*Csch[Pi] >> >> which agrees with the answer derived earlier by Boudewijn. I wonder what >> version of Mathematica you're using. > > OK, I've now tried Version 5.2 on three machines: two Mac OS X 10.4.5, > and one Windows XP. All give the same result. > > So 5.2 is a slight "downgrade". I'll add a followup to > comp.soft-sys.math.mathematica. > > Hello, Thank you for reporting the problem with the above infinite product. For this example, we first calculate the corresponding finite product which is easy since the input is a rational function. The bad result containing internal variables arises while taking the limit at Infinity using Series. We expect to fix this issue in a future release. A possible workaround for the problem is to use Limit as shown below to arrive at the result given in Mathematica 5.1. ========================================= In[1]:= $Version Out[1]= 5.2 for Linux (June 27, 2005) In[2]:= p = Product[(n^2 - 1)/(n^2 + 1), {n, 2, m}] Pi Csch[Pi] Gamma[m] Gamma[2 + m] Out[2]= ------------------------------------- Gamma[(1 - I) + m] Gamma[(1 + I) + m] In[3]:= p1 = Limit[p, m -> Infinity] Out[3]= Pi Csch[Pi] In[4]:= N[p1] Out[4]= 0.272029 In[5]:= NProduct[(n^2 - 1)/(n^2 + 1), {n, 2, Infinity}] Out[5]= 0.272029 + 0. I ======================================== I apologize for the inconvenience caused by this problem. Sincerely, Devendra Kapadia. Wolfram Research, Inc.

**References**:**Re: infinite product***From:*Ronald Bruck <bruck@math.usc.edu>