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Re: Incongruence? hmm...
*To*: mathgroup at smc.vnet.net
*Subject*: [mg102711] Re: [mg102710] Incongruence? hmm...
*From*: Leonid Shifrin <lshifr at gmail.com>
*Date*: Fri, 21 Aug 2009 04:41:53 -0400 (EDT)
*References*: <200908200856.EAA05738@smc.vnet.net>
Filippo,
First, the correct answer: Mathematica is right, given that 0<=x<=2Pi. Take
what Mathematica
gives if you only need positive x, or use a Sign[x] function in front of
cubic term, if you also need the interval -2Pi<=x<=0:
1/720 (8 \[Pi]^4 - 60 \[Pi]^2 x^2 + Sign[x]*60 \[Pi] x^3 - 15 x^4)
You should not have expanded the Cos, since then your sum over m
Sum[((m x)^(2n) (-1)^n)/((2n)!m^4),{m,1,Infinity}]
is not convergent for n>2. I am actually surprised that Mathematica did not
issue a warning when performing this sum. But in fact, it gave you a hint
that
the behavior changes qualitatively at n=2 - the argument of Zeta changes
sign.
So, the values of Zeta are indeed zero for n>2, but for n>2 this is no
longer
a correct answer - this is where the cubic term is coming from. The fact
that
there is a Sign function in front means that there is no way you could have
obtained this term through expansion in x - it is "non-perturbative". It is
another
way of saying that you should not have expanded.
One other way to obtain a correct answer valid also for negative x
(-2Pi<=x<=0)
is to ask Mathematica to compute
Sum[Sin[m x]/m^5, {m, 1, \[Infinity]}],
then differentiate over x and simplify the result under an assumption that x
is real.
Eventually, you will get the second result I mentioned. Needless to say, the
result is only
correct for -2Pi<=x<=2Pi.
Regards,
Leonid
On Thu, Aug 20, 2009 at 12:56 PM, Filippo Miatto <miatto at gmail.com> wrote:
> Dear all,
> I'm calculating the sum
>
> Sum[Cos[m x]/m^4, {m, 1, \[Infinity]}]
>
> in two different ways that do not coincide in result.
> If i expand the cosine in power series
>
> ((m x)^(2n) (-1)^n)/((2n)!m^4)
>
> and sum first on m i obtain
>
> ((-1)^n x^(2n) Zeta[4-2n])/(2n)!
>
> then I have to sum this result on n from 0 to infinity, but Zeta[4-2n]
> is different from 0 only for n=0,1,2 and the result is
>
> \[Pi]^4/90 - (\[Pi]^2 x^2)/12 - x^4/48
>
> Three terms, one independent on x, with x^2, one with x^4.
>
> however if I perform the sum straightforwardly (specifying that
> 0<x<2pi) the result that Mathematica gives me is
>
> \[Pi]^4/90 - (\[Pi]^2 x^2)/12 + (\[Pi] x^3)/12 - x^4/48
>
> with the extra term (\[Pi] x^3)/12. Any idea on where it comes from??
> Thank you in advance,
> Filippo
>
>
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