Hi, I got a question on the error bound.
lets say f(x) = (1)/(1-x) b=0 bound error (-1,1) and find where the error is <.001.
$\displaystyle 1+x+x^2+...+x^n = \frac{1 - x^{n+1}}{1 - x} = \frac{1}{1-x} - \frac{x^{n+1}}{1 - x}$
Note that $\displaystyle 1+x+x^2+...+x^n = T_n(x)$ - the $\displaystyle n$th Taylor polynomial.
Therefore, $\displaystyle \left| T_n(x) - \frac{1}{1-x} \right| = \frac{|x|^{n+1}}{|1-x|} $
The error term $\displaystyle \frac{|x|^{n+1}}{|1-x|}$ cannot be bounded uniformly.
Only for a particular value of $\displaystyle x\in (-1,1)$ can you make the error sufficiently small.