Find power a series representation for the functions

A) f(x)= 5/(1-4x^2)

B) f(x)= x/(1+4x)^2

please show steps thank you.

A) Start with the geometric series 1/(1 - t) = Σ(n = 0 to ∞) t^n.

Let t = 4x^2:
1/(1 - 4x^2) = Σ(n = 0 to ∞) (4x^2)^n
.................= Σ(n = 0 to ∞) 4^n x^(2n).

Multiply both sides by 5:
5/(1 - 4x^2) = Σ(n = 0 to ∞) 5 * 4^n x^(2n).
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B) Start with the geometric series 1/(1 - t) = Σ(n = 0 to ∞) t^n.

Differentiate both sides:
1/(1 - t)^2 = Σ(n = 1 to ∞) nt^(n-1).

Let t = -4x
1/(1 - (-4x))^2 = Σ(n = 1 to ∞) n(-4x)^(n-1)
==> 1/(1 + 4x)^2 = Σ(n = 1 to ∞) n (-4)^(n-1) x^(n-1)

Multiply both sides by x:
x/(1 + 4x)^2 = Σ(n = 1 to ∞) n (-4)^(n-1) x^n.
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I hope this helps!

(1+x)^n= 1+ nx+ n(n-1)x*x/2!+ (n(n-1)(n-2)x^3)/3!+............
for A)x=(-4X^2) n=(-1)
B)keep aside the numerator x and then n=(-2) and x=4X

A)
5 + 20 x^2 + 80 x^4 + 320 x^6 + 1280 x^8 + 5120 x^10

B)
x - 8 x^2 + 48 x^3 - 256 x^4 + 1280 x^5 - 6144 x^6 + 28672 x^7 - 131072 x^8 + 589824 x^9