Editing Radius of convergence (section)

==Rate of convergence==

If we expand the function

:<math>\sin x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\text{ for all } x</math>

around the point ''x'' = 0, we find out that the radius of convergence of this series is <math>\infty</math> meaning that this series converges for all complex numbers.  However, in applications, one is often interested in the precision of a [[numerical analysis|numerical answer]].  Both the number of terms and the value at which the series is to be evaluated affect the accuracy of the answer.  For example, if we want to calculate {{math|sin(0.1)}} accurate up to five decimal places, we only need the first two terms of the series.  However, if we want the same precision for {{math|1=''x'' = 1}} we must evaluate and sum the first five terms of the series.  For {{math|sin(10)}}, one requires the first 18 terms of the series, and for {{math|sin(100)}} we need to evaluate the first 141 terms.

So for these particular values the fastest convergence of a power series expansion is at the center, and as one moves away from the center of convergence, the [[rate of convergence]] slows down until you reach the boundary (if it exists) and cross over, in which case the [[Series (mathematics)|series]] will diverge.