Taylor series based on power series

Question

Set up the Taylor series to $f (x) = \ln (1 + 2x)$ about$ x = 0 $ based on the power series $ \frac{1}{1-x}$ . I have found that Taylor series of $f(x)$ is $2x-2x^2+\frac{8}{3}x^3-4x^4+...,$ but how do I base it on $\frac{1}{1-x}$?

Answer 1

Hint. Note that $$\ln(1+2x)=-\int_0^{-2x}\frac{1}{1-t}\,dt$$