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Re: Incongruence? hmm...

  • To: mathgroup at smc.vnet.net
  • Subject: [mg102717] Re: [mg102710] Incongruence? hmm...
  • From: "Tony Harker" <a.harker at ucl.ac.uk>
  • Date: Fri, 21 Aug 2009 04:43:01 -0400 (EDT)
  • References: <200908200856.EAA05738@smc.vnet.net>

 I don't see how either expression can be correct for all x. The 'direct'
form is clearly wrong, as every term in the summation is an even function of
x. However, the modulus of each term is less than or equal to 1/m^4, so the
sum must be bounded above and below by plus and minus Pi^4/90, which neither
result is.

  The indirect approach obviously has a limited radius of convergence for
the sum over m, and GenerateConditions->True will show that, so that's what
goes wrong there. The direct approach does not generate any conditions, so
seems to be just plain wrong. 

  The key to sorting this out is to note that the original expression is a
Fourier series, so any polynomial form can only be valid over an interval
such as -Pi to Pi, and must then repeat periodically.

  In fact, the result from the direct sum is correct for 0<x<Pi, and for
-Pi<x<0 the expression is similar, but the sign of the coefficient of x^3 is
changed. Basically, the original sum is a polynomial in (Pi-x) which has
been made symmetrical about x=0.

  Tony Harker


]-> To: mathgroup at smc.vnet.net
]-> Subject: [mg102710] Incongruence? hmm...
]-> 
]-> 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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