TaylorAndMaclaurin

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Author:
sublime1
ID:
48653
Filename:
TaylorAndMaclaurin
Updated:
2010-11-10 02:58:39
Tags:
Taylor Maclaurin Series Formulas Math
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Description:
Taylor and Maclaurin Series Formulas - Math 113
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  1. Theorem 5
    • If f has a power series representation (expansion) at a, that is, if: f(x) = cn(x - a)n | x - a| < R THEN its coefficients are given by the formula: cn = f(n)(a) / n!
  2. Formula 6 - Taylor series of the function f at a
    • f(x) = ( f(n)(a) / n! ) (x - a)n
    • = f(a) + ( f ' (a) / 1! ) ( x - a) + ( f '' (a) / 2! ) ( x - a)2 + ( f ''' (a) / 3! ) ( x - a)3 + ...
  3. Formula 7 - Taylor series of the function f at 0 (Maclaurin series)
    • f(x) = ( f(n)(0) / n! ) xn
    • = f(0) + ( f ' (0) / 1! ) x + ( f '' (0) / 2! ) ( x)2 + ( f ''' (0) / 3! ) ( x)3 + ...
  4. Theorem 8
    if f(x) = Tn(x) + Rn(x), where Tn is the nth-degree Taylor polynomial of f at a and lim(n->infinity) Rn(x) = 0 for |x-a| < R, the f is equal to the sum of its Taylor series on the interval |x-a| < R

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