I have to prove that at least seven terms must be used in the Taylor series estimation of x - sin(x) in order for the error to be <= $10^{-9}$. This doesn't seem correct however. This series is equal to the summation of $\frac{-1^n * x^{2n+1}}{(2n+1)!}$ and the error term that I get for 1n = 7 doesn't even come close to $10^{-9}$
edit $0 < x < 1$
Since $0<x<1$, it is simplest to use the Alternating Series Test estimate for the error (although Taylor's Remainder Formula will give the same estimate):
If we use the first 5 nonzero terms of the series,
$\;\;\;\displaystyle\frac{x^3}{3!}-\frac{x^5}{5!}+\frac{x^7}{7!}-\frac{x^9}{9!}+\frac{x^{11}}{11!}$,
the error satisfies $|E|<\frac{1}{13!}<10^{-9}$ since $13!>10^9$.
(Notice that $\frac{1}{11!}>10^{-9}$, so we need at least 5 nonzero terms.)