Taylor Polynomials,Planck's Law and Rayleigh-Jeans Law
Use a taylor polynomial to show that , for a large wavelength, Planck's Law gives approximately the same values as the Rayleigh-Jeans Law. see the attached problem #2© SolutionLibrary Inc. solutionlibary.com 9836dcf9d7 https://solutionlibrary.com/math/algebra/taylor-polynomials-plancks-law-and-rayleigh-jeans-law-2l7
...lynomials for functions that we already know. I would start with that exponential. Now for when x is small ( x ~ 0)
exp(x) = 1 + x + x^2/2 + x^3/6 + ... + x^n/n! + ...
In our case, the x is hc/(lambda K T). When lambda is large this fraction is small and thus close to zero - so the approximation using the Taylor polynomial applies. Substitute this into the function:
f(lambda) ~ 8 Pi h c lambda^(-5) / ( 1 + (hc/lambda k T) + (hc/lambda k T)^2/2 + ... - 1)
See how I just replaces exp() with its Taylor polynomial? Notice I replaced the = (equals) symbol with a ~ to mean 'approximately equal to'.
Now we can do some ...