TAYLOR SERIES BY:DUBAN CASTRO FLOREZ NUMERICS METHOS IN ENGINEERING PETROLEUM ENGINEERING 2010 CONSTRUIMOS FUTURO 1
TAYLOR SERIES In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. If the series is centered at zero, the series is also called a Maclaurin series. It is common practice to use a finite number of terms of the series to approximate a function. The Taylor series may be regarded as the limit of the Taylor polynomials. CONSTRUIMOS FUTURO 2
CONSTRUIMOS FUTURO 3 TAYLOR SERIES The Taylor series of a real or complex function ƒ(x) that is infinitely differentiable in a neighborhood of a real or complex number a is the power series: which can be written in the more compact sigma notation as: where n! denotes the factorial of n and ƒ (n)(a) denotes the nth derivative of ƒ evaluated at the point a. The zeroth derivative of ƒ is defined to be ƒ itself and (x − a)0and 0! are both defined to be 1.
4 TAYLOR SERIES orderzero orderone ordertwo order n CONSTRUIMOS FUTURO
CONSTRUIMOS FUTURO 5 MACLAURIN SERIES In the particular case wherea = 0, the series isalsocalled a Maclaurin series:
6 EXAMPLE For f(x) = arccos (x) a) to Write the polynomial of Mclaurin P3(x) for f(x). CONSTRUIMOS FUTURO
CONSTRUIMOS FUTURO 7 b) to Complete the following chart for P3(x) and it stops f(x) (to Use radianes).