A physicist models the spectral density S(λ)S(\lambda)S(λ) of a laser beam for small wavelengths λ \lambda\,λ using the function
S(λ)=(14−12λ)−52∣λ∣<12 S(\lambda) = \left( \frac{1}{4} - \frac{1}{2}\lambda \right)^{-\frac{5}{2}} \quad |\lambda| < \frac{1}{2} S(λ)=(41−21λ)−25∣λ∣<21Determine the first 4 terms of the binomial expansion of S(λ)S(\lambda)S(λ) in ascending powers of λ\lambdaλ, simplifying each coefficient.
A secondary filter is applied such that the resulting density R(λ)R(\lambda)R(λ) satisfies
(14−12λ)nS(λ)=(14−12λ)−12 \left( \frac{1}{4} - \frac{1}{2}\lambda \right)^{n} S(\lambda) = \left( \frac{1}{4} - \frac{1}{2}\lambda \right)^{-\frac{1}{2}} (41−21λ)nS(λ)=(41−21λ)−21State the value of nnn.
Hence, or otherwise, find the first 3 terms of the binomial expansion of R(λ)R(\lambda)R(λ) in ascending powers of λ\lambdaλ, giving each term in its simplest form.