kde2520
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Homework Statement
The Bloch-Gruneissen approximation for the resistance on a monovalent metal is
\rho=C(T^{5}/\Theta^{6})\int^{\Theta/T}_{0}\frac{x^{5}dx}{(e^{x}-1)(1-e^{-x})}
(a)For T->\infty, show that \rho=(C/4)(T/\Theta^{2})
(b)For T->0, show that \rho=5!\zeta(5)C\frac{T^{5}}{\Theta^{6}}
Homework Equations
The section is on Bernoulli numbers and the Euler-Maclaurin Formula. Several definitions including x/(e^x-1)=sum->(Bn*x^n)/n!, Bernoulli Polynomials, Reimann-Zeta function, etc.
The Attempt at a Solution
For part (a) I see that as T->infinity the upper integration limit goes to zero, thus I may approximate the integrand giving (as the integrand) x^5/[(x+x^2/2!+x^3/3!+...)(-x+x^/2!-x^3/3!+...)]. Can I just multiply this out, simplify, and integrate term by term? If so, over what integration limits?
For part (b) the upper limit goes to infinity so I'm guessing I need to do the integral by substituting some definition of the Bernoulli Numbers?...
Help...
PS - Sorry if the equations are unclear. I'm new to LaTex. Help with that would be appreciated too.