テイラー展開 - math.chs.nihon-u.ac.jp

1 . f (x) x = a . f 00 (a). f (x) = f (a) + f 0 (a)(x a) + (x a)2 + . 2.. |x a| < a ( . ) > 0 f x = a . f . (C ) a = 0 .. 2 .. 1 f 0 (C 1 ) .. f (x) f (a). = f 0 (c) (c x a ). x a . f (x) = f (a) + f 0 (c)(x a). c = a + (x a) (0 1) . f (x) = f (a) + f 0 (a + (x a))(x a) (0 1). x a .. 1. n n f (n) (C n ) . f 00 (a) f (n 1) (a). f (x) = f (a) + f 0 (a)(x a) + (x a)2 + + (x a)n 1. 2 (n 1)! f (n) (a + (x a)). + (x a)n n! . f (n) (a + (x a)). Rn = (x a)n n! . n 1 (n 1 ). f 00 (a) f (n 1) (a). f 0 (a)(x a) + (x a)2 + + (x a)n 1. 2 (n 1)! f (x) Rn .. sin x . x3 x5. sin x = x + + . 3! 5! . f (n) ( x) n Rn = x n! f (n) ( x) sin( x) cos( x) 1 .. |x|n |Rn | . n! . sin x . 2 (0 . ) 2 3 (1 2 ) sin x.

テイラー展開 森 真 1 序 関数f(x) のx = a におけるテイラー展開は f(x) = f(a)+f0(a)(x−a)+ f00(a) 2 (x−a)2 +···と表される．右辺の級数が収束する範囲でのみ成り立つことに注意しなけれ

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