I posted this task yesterday. I got help with understanding how to solve a part of the problem but the other part still confuses me. The task is:
Use the power series to $f(x)=\frac{1}{1-x}$ to find the Taylor series to
$g(x)=\ln(1+2x)$ about $x=0$
I then did this
$$\int\frac{1}{1-x}=\int\sum{x^n}$$ $$-\ln(1-x)=\sum{\frac{x^{n+1}}{n+1}}$$ $$\ln(1-x)=-\sum{\frac{x^{n+1}}{n+1}}$$
Then I use substituion and plugg in $-2x$ for $x$ and get
$$\ln(1+2x)=\sum{\frac{(-1)^{n+2}(2x)^{n+1}}{n+1}}$$
So the question remaining is: How many terms should one have in the estimation of $\ln(1,02)$ for the error to be less than $2\times10^{-6}$?. I don't know where to begin. Should I use the Langrange remainder? I don't understand where to begin. Really appreciate some help.