I'm asked to find the Taylor-series to
$f(x)=\ln(1+2x)$ about $x=0$ using the power series to $\frac{1}{1-x}$. And then I must find the biggest number of $n$ necessary to estimate $\ln(1.02),$ where the biggest error can be $2.0 \times 10^{-6}.$
So I think I must integrate
$\int{\frac{1}{1-x}}dx=-\ln(1-x)=\sum \frac{x^{n+1}}{n+1}$
And then I'm stuck. I really don't understand how to use that power serie to find the Taylor series.
Best Answer
Use the power series $\frac1{1-t} = 1 + t + t^2 + t^3+...$ to express
$$\ln(1-y) = -\int_0^y \frac1{1-t} dt = -\int_0^y ( 1 + t + t^2 + ...) dt =-y-\frac12 y^2 - \frac 13 y^3+...$$
Then, set $y = -2x$ to get the series for $\ln(1+2x)$, $$\ln(1+2x) = 2x-2x^2+\frac83 x^3+...$$
Plug in $x=0.01$ to approximate $\ln(1.02)$.