Mathematical Methods for Physics and Engineering - Matematica.NET

SERIES AND LIMITSvalue of ξ that satisfies the expression for R n (x) is not known, an upper limit onthe error may be found by differentiating R n (x) withrespecttoξ and equatingthe derivative to zero in the usual way for finding maxima.◮Expand f(x) =cosx as a Taylor series about x =0and find the error associated withusing the approximation to evaluate cos(0.5) if only the first two non-vanishing terms aretaken. (Note that the Taylor expansions of trigonometric functions are only valid for anglesmeasured in radians.)Evaluating the function and its derivatives at x = 0, we findf(0) = cos 0 = 1,f ′ (0) = − sin 0 = 0,f ′′ (0) = − cos 0 = −1,f ′′′ (0) = sin 0 = 0.So, for small |x|, we find from (4.18)cos x ≈ 1 − x22 .Note that since cos x is an even function, its power series expansion contains only evenpowers of x. Therefore, in order to estimate the error in this approximation, we mustconsider the term in x 4 , which is the next in the series. The required derivative is f (4) (x)and this is (by chance) equal to cos x. Thus, adding in the remainder term R 4 (x), we findcos x =1− x22 + x4cos ξ,4!where ξ lies in the range [0,x]. Thus, the maximum possible error is x 4 /4!, since cos ξcannot exceed unity. If x =0.5, taking just the first two terms yields cos(0.5) ≈ 0.875 witha predicted error of less than 0.00260. In fact cos(0.5) = 0.87758 to 5 decimal places. Thus,to this accuracy, the true error is 0.00258, an error of about 0.3%. ◭4.6.3 Standard Maclaurin seriesIt is often useful to have a readily available table of Maclaurin series for standardelementary functions, and therefore these are listed below.sin x = x − x33! + x55! − x7+ ···7!for −∞