When we consider the error in a Taylor polynomial (finite) approximation, there are two factors to consider:[br][list][*]the degree of the polynomial we will use (higher degree is more accurate, but takes longer to compute), and[/*][*]the specific x value where want to approximate sin(x); closer to the x=a center is better, further out the approximation will get worse. [/*][/list][br]The bound on the Taylor Polynomial error can be computed in one of two ways:[list][*]If the series is alternating for the given x value, we can use the Alternating Series Error Bound of the first omitted term.[/*][*]If the series is [i]not[/i] alternating for the given x value, we have to use Taylor's inequality, which is a little more complicated.[/*][/list][br]For the power series for [math]f\left(x\right)=x^{\frac{\frac{1}{3}}{ }}[/math], the series is alternating when x > 8, but not when x < 8. That means[br]- for x>8, the first omitted term is large enough error to include the original function, but[br]- for x<8, the first omitted term is [i]not[/i] large enough error, and we need to use the larger Taylor Inequality error.