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III. Potensrækker – 3. Potensrækker 294 Substitueres −x 2 for x i eksponentialrækken, fås eller (for alle x). e −x2 = 1 − x 2 + 1 2! x4 − 1 3! x6 + ... e −x2 ∞ (−1) = n n=0 n! x2n 3.30. Gauss’ fejlintegral ☞ [S] 8.7 Taylor and Maclaurin series Eksempel 8 - fortsat For f(x) = e−x2 dx (med f(0) = 0) er en rækkeudvikling med centrum i 0 e −x2 ∞ (−1) dx = n e −x2 ∞ dx = n=0 n=0 n! x2n (−1) n (2n + 1)n! x2n+1 3.31. Gauss’ fejlintegral (fortsat) Eksempel 8 - fortsat e −x2 dx = x − x3 x5 x7 x9 + − + − ... 3 · 1! 5 · 2! 7 · 3! 9 · 4!
III. Potensrækker – 3. Potensrækker 295 1 Sum af de anførte led, sand værdi 0.746824... 0 e −x2 dx = x − x3 1 x5 x7 x9 + − + − ... 3 · 1! 5 · 2! 7 · 3! 9 · 4! 0 = 1 − 1 1 1 1 + − + − ... 3 · 1! 5 · 2! 7 · 3! 9 · 4! 0.747487... 3.32. Opgave ☞ Matematik Alfa 1, August 2002 Opgave 4 Angiv en potensrække i x, der for x = 0 fremstiller funktionen Angiv også grænseværdien f(x) = cos(x2 ) − 1 x 4 lim x→0 f(x). 3.33. Opgave ☞ Matematik Alfa 1, August 2002 Opgave 4 - Løsning cos x = 1 − x2 2! + x4 4! − ...
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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3. Generelle områder 201 4. Koordi
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