\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
For the function \(f(x)=\ln(1-x)\),
  1. Find \(f^{(1)}(x)\), \(f^{(2)}(x)\), and \(f^{(3)}(x)\).
  2. Find \(f(0)\), \(f'(0)\), \(f^{(2)}(0)\), and \(f^{(3)}(0)\).
  3. Show that the Maclaurin series for \(\ln(1-x)\) is$$ \ln(1-x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - \dots = -\sum_{k=1}^\infty \frac{x^k}{k} $$

Capture an image of your work. AI teacher feedback takes approximately 10 seconds.