Suppose we have this simple rational function and the expansion of it in few terms at the origin,
$$\frac{1}{1-x^4}= 1 + x^4 + \, ...$$
Now what happen if we want to compare the error between $n=2$ and $n=3$? It seems as if, when we are using Lagrange's Remainder, the error is actually reduced. Because the denominator increases with greater $n$. But we know even for $n=3$, we only have two terms. Can anyone spot where my misconception is? Or if this is not actually a problem because for $n < 4$, the error is always taken as the smallest one anyway i.e the error for $n=2$ and $n=3$ is always $R_3$? (Where $R_n$ is the Lagrange's Remainder).
Thank You
When we use the formula for $n = 3$, we're using more information about the function than we do for $n = 2$, so it's no surprise that we're able to extract a tighter bound on the error in the former case.