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Rate of convergence

If we expand the function

around the point x = 0, we find out that the radius of convergence of this series is ∞ {\displaystyle \infty } meaning that this series converges for all complex numbers. However, in applications, one is often interested in the precision of a 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 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 x = 1 we must evaluate and sum the first five terms of the series. For sin(10), one requires the first 18 terms of the series, and for 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 will diverge.