If f(x)=∑n=0∞xn=11−xf(x) = \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}f(x)=∑n=0∞xn=1−x1 and g(x)=∑n=0∞(n+1)xng(x) = \sum_{n=0}^{\infty} (n+1)x^ng(x)=∑n=0∞(n+1)xn (both for ∣x∣<1|x| < 1∣x∣<1), which relationship holds?
g(x)=f′(x)g(x) = f'(x)g(x)=f′(x)
g(x)=ddx[xf(x)]g(x) = \frac{d}{dx}[xf(x)]g(x)=dxd[xf(x)]
g(x)=xf′(x)g(x) = xf'(x)g(x)=xf′(x)
g(x)=(1−x)f(x)g(x) = (1-x)f(x)g(x)=(1−x)f(x)