Question

How to find the Maclaurin series for f(x) using definition of a Maclaurin series: f(x) = ln(1+x)

Answer 1

\[f'(x)=\frac{1}{1+x},f'(0)=\frac{1}{1}=1\]\[f''(x)=\frac{-1}{(1+x)^2},f''(0)=\frac{-1}{1^2}=-1\]\[f'''(x)=\frac{2}{(1+x)^3},f''(0)=\frac{2}{1^3}=2\]

Answer 2

Generally, for n>1, \[f^n(0)=(-1)^{n-1}(n-1)!\]

Answer 3

Now just use the definition of Maclaurin:
http://mathworld.wolfram.com/MaclaurinSeries.html

Specifically, http://mathworld.wolfram.com/images/equations/MaclaurinSeries/NumberedEquation1.gif

Answer 4

Any problem?