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III. Potensrækker – 3. Potensrækker 292 En vilkårlig ofte differentiabel funktion f(x) har Taylorrække om a ∞ 6 f(x) = og Maclaurinrække, a = 0, n=0 7 f(x) = f (n) (a) (x − a) n! n ∞ n=0 f (n) (0) x n! n 3.26. Eksponentialrækken som Maclaurin række ☞ [S] 8.7 Taylor . . . Eksempel 1 For f(x) = e x er f (n) (x) = e x for alle n. Så er f (n) (0) = e 0 = 1 for alle n, så Maclaurin rækken for e x er e x = ∞ n=0 1 n! xn 3.27. Sinusrække som Maclaurin række ☞ [S] 8.7 Taylor . . . Eksempel 4 For f(x) = sin x er sin ′ x = cos x og cos ′ x = −sin x. Så f(0) = 0,f ′ (0) = 1, f ′′ (0) = 0,f ′′′ (0) = −1,
III. Potensrækker – 3. Potensrækker 293 Maclaurin rækken er 15 sinx = x − x3 3! f 4 (0) = 0 0,1,0, −1,0,1,0, −1,0,1,0, −1,0,1,... + x5 5! − x7 7! + ... 3.28. Cosinusrække som Maclaurin række ☞ [S] 8.7 Taylor . . . Eksempel 5 For f(x) = cos x, f(0) = 1,f ′ (0) = 0,f ′′ (0) = −1,f ′′′ (0) = 0,... Maclaurin rækken er 1,0, −1,0,1,0, −1,0,1,0, −1,... cos x = 1 − x2 x4 + − ... 2! 4! Denne rækkeudvikling kan også udledes ved at differentiere sinx = x − x3 3! + x5 5! − ... 3.29. Gauss’ fejlintegral ☞ [S] 8.7 Taylor and Maclaurin series Eksempel 8
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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