Bernoulli Numbers, Euler-Maclauring Formula - Math Methods class

[kde2520]

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.

 

[Avodyne]

Can I just multiply this out, simplify, and integrate term by term?

Yep! (You have a sign wrong though ...)

If so, over what integration limits?

Same as before, you just think of the upper limit as a small parameter.

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?...

Try expanding everything but the $x^5$ in powers of $e^{-x}$.

As for LaTeX, just put the beginning and ending tex and /tex commands around the whole equation:

$$\rho=C(T^{5}/\Theta^{6})\int^{\Theta/T}_{0}\frac{x^{5}dx}{(e^{x}-1)(1-e^{-x})}$$

 

[kde2520]

I guess you mean the sign is wrong in the expansion of $$e^{-x}$$? What is it's expansion?

 

[kde2520]

Try expanding everything but the $x^5$ in powers of $e^{-x}$.

Isn't that what we did for part (a)?

Thanks for the LaTex tip!

 

[Avodyne]

I mean an expansion like

$${1\over 1-e^{-x}}=\sum_{n=0}^\infty (e^{-x})^n$$