Find a power series representation for the function ....?

Start with the geometric series

1/(1 - t) = Σ(n = 0 to ∞) t^n, convergent for |t| < 1.

Differentiate both sides:

1/(1 - t)^2 = Σ(n = 1 to ∞) nt^(n-1), still convergent for |t| < 1.

Let t = -3x^2:

1/(1 - (-3x^2)^2 = Σ(n = 1 to ∞) n(-3x^2)^(n-1), convergent for |-3x^2| = 3|x|^2 < 1.

==> 1/(1 + 3x^2)^2 = Σ(n = 1 to ∞) n(-3)^(n-1)x^(2n-2), convergent for |x| < 1/√3.

Multiply both sides by x^3:

x^3/(1 + 3x^2)^2 = Σ(n = 1 to ∞) n(-3)^(n-1)x^(2n+1), convergent for |x| < 1/√3.

(radius of convergence is 1/√3)

I hope this helps!

let f1 = 1 / (1 +u)^2

find series of f1 then substitute u = 3x^2

series converges when 3x^2 <1 or

- sqrt(3)/3 < x < sqrt(3) /3