So I am aware that with the increase in number of terms the approximation becomes more accurate but I wished to know if I have a binomial of the form say,
$$ f(x) = (1-Rx)^{-n}$$
where $R$ is a real number and $n$ is a natural number, then is there a relation using which I can determine the accuracy of my approximation. Say, the above $f(x)$ is expanded as below,
$$f(x) = 1 + ax + bx^2 + cx^3 + ... $$ $$b(x) = 1 + ax + bx^2$$
then, does there exist some formula or relation using which I can conclude that if I take the first three terms of $f(x)$ and find a value at $1/10R$, then, $b(1/10R)$ will only be accurate upto the first decimal value of $f(1/10R)$.
I have tried it on some integer values and that seemed to be the case but I think it is just the bias in my data. So, I would like to know if there indeed exists some formula or intuitive way to calculate the accuracy of the approximation.
I also posted the question on MO:- https://mathoverflow.net/questions/448831/is-there-a-formulaic-relation-between-the-accuracy-of-the-approximation-and-the