标题: 数学名词站 [打印本页] 作者: lilianjie 时间: 2011-12-27 16:37 标题: 数学名词站 本帖最后由 lilianjie 于 2011-12-27 16:39 编辑 * k5 f$ T1 _. |9 k" }% H7 I
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cut-the-knot。ORG & A2 k+ M5 S9 |3 }% Y( s& R* ^* C0 m6 C' v7 p
Maclaurin and Taylor series ; ~; L1 l( u; a! x/ m1 U+ n3 kFor a (real) function f under certain conditions (Taylor's Theorem) " S5 x5 a: p/ f! x7 \0 H: e, G K + `- _! d& k9 A4 g h) F f(x) = f(a) + (x - a)f'(a) + (x - a)2f(2)(a)/2! + ... + (x - a)nf(n)(a)/n! + Rn / e) _; y7 L; J( F$ K# s7 d
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One obtains a Maclaurin series when a = 0. However, introducing g(x) = f(x + a) one gets f(n)(a) = g(n)(0), and so the Maclaurin series for g at x = 0 coincides with the Taylor series of f at x = a./ G6 G4 _& j( `) H
& D# H: E M- eThe remainder Rn looks very much like the expected next term, with the derivative evaluated at an intermediate point:1 B8 J, h, p; j& j( a( q- r
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Rn = (x - a)n + 1f(n + 1)(γ)/(n + 1)!, . X M4 E) K/ l J) a) T& e% _! b ~. X5 |, u! D9 O
where γ is a point between a and x. For the derivation of this form for the remainder of the series f is required to have at least n + 1 continuous derivatives.! }. C+ A1 g4 M/ U: _$ M