On Dec 11, 10:59 pm, achrzesz <achrz...@wp.pl> wrote:
> On Dec 11, 7:30 pm, achrzesz <achrz...@wp.pl> wrote:
>
>
>
> > sage: N(pi^2/6, digits=17)
> > 1.6449340668482264
> > sage: numerical_integral(x/(exp(x)-1),0,oo)
> > (1.6449340668482264, 5.9356452836178026e-10)
>
> > On Dec 11, 6:45 pm, "andres.ordonez" <andres.felipe.ordo...@gmail.com>
> > wrote:
>
> > > That'll do. Thanks
>
> > > On Dec 10, 8:19 pm, Renan Birck Pinheiro <renan.ee.u...@gmail.com>
> > > wrote:
>
> > > > 2011/12/10 andres.ordonez <andres.felipe.ordo...@gmail.com>
>
> > > > > Hi, I'm having trouble evaluating this integral
>
> > > > > integral( x / (exp(x) - 1) , (x,0,oo)).n()
>
> > > > > I get
>
> > > > > TypeError: cannot evaluate symbolic expression numerically
>
> > > > > The answer (according to mathematica) should be pi^2 / 6
>
> > > > > Is something wrong with my code?
>
> > > > > Thanks!
>
> > > > Apparently it stumbles upon a limit which Maxima is incapable of doing.
>
> > > > sage: integrate(x/(exp(x)-1),(x,0,oo))
> > > > -1/6*pi^2 + limit(-1/2*x^2 + x*log(-e^x + 1) + polylog(2, e^x), x,
> > > > +Infinity)
>
> > > > if oo is replaced by a very large number, seems to work...
>
> > > > sage: (real_part(integrate(x/(exp(x)-1),(x,0,1000)).n()) - (pi^2/6)).n()
> > > > -4.41002789841605e-11
>
> > > > --
> > > > Renan Birck Pinheiro - Grupo de Microeletrônica
> > > > <http://www.ufsm.br/gmicro>- Engenharia
> > > > Elétrica <http://www.ufsm.br/cee>/UFSM <http://www.ufsm.br>
>
> > > >http://renanbirck.blogspot.com/skype:renan.ee.ufsm
>
> Using the geometric series one can obtain
>
> x/(e^x-1)=xe^(-x)/(1-e^(-x))=\sum_0^\infty x(e^(-(k+1)x)
>
> Integrating term by term
>
> sage: var('k x')
> sage: maxima('assume(k>-1)')
> [k>-1]
> sage: maxima('integrate(x*exp(-(k+1)*x),x,0,inf)')
> 1/(k+1)^2
>
> one can obtain the exact integral
>
> sage: sum(1/k^2,k,1,oo)
> 1/6*pi^2
>
> Andrzej Chrzeszczyk
Maxima knows the expansion:
sage: maxima('powerseries(x*exp(-x)/(1-exp(-x)),exp(-x),0)')
x*%e^-x*'sum(%e^-(i1*x),i1,0,inf)
Andrzej Chrzeszczyk
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