The Taylor [i]Series [/i]for e(x) expanded around x=0 is[br][math]e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...[/math] [br]which is an infinite series.[br]If we approximate e^x with a [i]finite [/i]portion of that series, we are using a Taylor [i]Polynomial.[br][br][/i]When we consider the error in that 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=0 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:[br][br][br][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]