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III. Potensrækker – 3. Potensrækker 288 Eksempel 6 - fortsat −ln(1 − x) = x + x2 2 + x3 3 −ln(1 − (1 − z)) = (1 − z) + + x4 4 (1 − z)2 2 eller (z − 1)2 lnz = (z − 1) − + 2 (z − 1)3 (substituer 1 − z for x; gælder for 0 < z ≤ 2). ...for − 1 < x < 1 + (1 − z)3 3 3 − ... + ... 3.18. Arctan rækken ☞ [S] 8.6 Representations of functions . . . Eksempel 7 For |x| < 1 er | − x 2 | < 1, så for sådanne x fås ved substitution i den geometriske række Integreres ledvis fås 1 1 + x 2 = 1 − x2 + x 4 − x 6 + ... Arctan(x) = x − x3 3 + x5 5 − ... 3.19. Gentagen differentiation ☞ [S] 8.7 Taylor and Maclaurin series Udregning f(x) = c0 + c1x + c2x 2 + c3x 3 + c4x 4 + ...
III. Potensrækker – 3. Potensrækker 289 f ′ (x) = c1 + 2c2x + 3c3x 2 + 4c4x 3 + ... f ′′ (x) = 2 · c2 + 3 · 2 · c3x + 4 · 3 · c4x 2 + ... f ′′′ (x) = 3 · 2 · 1 · c3 + 4 · 3 · 2 · c4x + ... f (4) (x) = 4 · 3 · 2 · 1 · c4 + 5 · 4 · 3 · 2 · c5x + ... 3.20. Gentagen differentiation ☞ [S] 8.7 Taylor and Maclaurin series Udregning - fortsat Indsættes x = 0, fås f(0) = c0, f ′ (0) = c1, generelt eller f ′′ (0) = 2 · c2, f ′′′ (0) = 3 · 2 · c3, f (4) (0) = 4 · 3 · 2 · c4,... f (n) (0) = n · (n − 1) · ... · 2 · 1 · cn f (n) (0) = n! · cn
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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3. Generelle områder 201 4. Koordi
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