Find the Binomial Series

Question

Find the Binomial Series of g(x) = 1/(1+5x^2)⁴

Answer 1

 

g(x) = (1+5x²)^-4

g(x) = ∑{k=0, ∞}C(α, k) (5x²)^k, where α = -4, and

 

C(α, k) = α(α-1)(α-2)...(α-k+1)/k!

Answer 2

The binomial series is

(1+u)ⁿ = ∑_k=1^∞ (n k) u^k   for |u|<1,

where (n k) is the generalized binomial coefficient, n(n-1)(n-2)...(n-k+1)/k!

 

Therefore,

(1+5x²)^-4 = ∑_k=1^∞ (-4 k) 5^kx^2k, where (-4 k) = (-4)(-5)(-6)...(-3-k)/k!

 

In particular, (-4 1) = -4, (-4 2) = (-4)(-5)/2!=10, (-4 3) = (-4)(-5)(-6)/3! = -20, so that the first three terms are:

(1+5x²)^-4 = -4 (5)x² +10(25)x⁴ -20(125)x⁶ +- ... = -20x² + 250x⁴ - 2500x⁶ +- ...