Taylor series for f(x)=ln(1+4x) centered about x=0

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

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

Integrate both sides from 0 to t:
ln(1 + t) = Σ(n=0 to ∞) (-1)^n t^(n+1)/(n+1).

Replace t with 4x:
ln(1 + 4x) = Σ(n=0 to ∞) (-1)^n 4^(n+1) x^(n+1)/(n+1)
...............= Σ(k=1 to ∞) (-1)^(k-1) 4^k x^k/k

I hope this helps!