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III. Potensrækker – 3. Potensrækker 282 For alle x med −1 ≤ x ≤ 1 gælder Arctan x = x − x3 x5 + − ... 3 5 (En syvende berømt række er binomialrækken, [S] 8.8.) 3.5. En potensrække ☞ [S] 8.5 Power series Definition En potensrække med centrum i a er et udtryk af form cn’erne kaldes rækkens koefficienter. Skrives også 2 Bemærk c0(x − a) 0 = c0, da (x − a) 0 = 1. c0 + c1(x − a) 1 + c2(x − a) 2 + c3(x − a) 3 + ... ∞ cn(x − a) n n=0 3.6. Logaritmerækken ☞ [S] 8.5 Power series Eksempel Logaritmerækken lnx = (x − 1) − er en potensrække med centrum i a = 1. (x − 1)2 2 + (x − 1)3 3 − ...
III. Potensrækker – 3. Potensrækker 283 Koefficienterne er c0 = 0,c1 = 1,c2 = − 1 2 ,c3 = 1 3 ,... 3.7. Eksponentialrækken ☞ [S] 8.5 Power series Eksempel Eksponentialrækken e x = 1 + x x2 x3 + + + ... 1! er en potensrække med centrum i a = 0. Koefficienterne er c0 = 1,c1 = 1/1!,c2 = 1/2!,c3 = 1/3!,... 3.8. Konvergens ☞ [S] 8.5 Power series 3 Sætning For en potensrœkke med centrum i a ∞ cn(x − a) n er der netop 3 muligheder (i) Konvergerer kun for x = a (ii) Konvergerer for alle x n=0 2! (iii) Der findes et tal R > 0 så rœkken er konvergent for |x−a| < R og divergent for |x−a| > R 3!
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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3. Generelle områder 201 4. Koordi
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