http://www.ck12.org Chapter 8. Infinite Series
- f(x) =lnxatx= 1 ,n= 4
- f(x) = 1 +x+x^2 +x^3 +x^4 atx=− 1 ,n= 4
Taylor and Maclaurin Series
Definition
(Taylor Series off)
The Taylor series of a functionfatx=x 0 is the power seriesT(x) =∞
n∑= 0 f(n)(x 0 )
n! (x−x^0 )n=f(x 0 )+f′(x 0 )(x−x 0 )+f′′(x 0 )
2! (x−x^0 )(^2) +f′′′(x^0 )
3! (x−x^0 )
(^3) +...
taking all the terms of the Taylor polynomials. The Maclaurin seriesM(x)offis the Taylor series atx=0.
Example 1Find the Maclaurin series off(x) =cosx.
Solution.f′(x) =−sinx, f′(x) =−cosx,f′′(x) =sinx,f′′′(x) =cosx, f(^4 )(x) =−sinx, f(^5 )(x) =−cosx,...
Notice the pattern repeats every 4 terms. Sof′( 0 ) = 0 ,f′( 0 ) =− 1 ,f′′( 0 ) = 0 ,f′′′( 0 ) = 1 ,f(^4 )( 0 ) = 0 ,x,f(^5 )( 0 ) =
− 1 ...
The Maclaurin series off(x) =cosxis
M(x) = 1 −2!^1 x^2 +4!^1 x^4 −6!^1 x^6 +8!^1 x^8 −...=∑∞n= 0 ((− 2 n^1 ))!nx^2 n.
ExercisesFind the Taylor series of following functions at the givenx 0.
- f(x) = 1 −^1 xatx= 0
- f(x) =exatx= 1
- f(x) =^1 xatx= 2
Convergence of Taylor and Maclaurin Series
Sincef(n)(x)is defined for all functionsfin this text, the Taylor seriesT(x)offis always defined. As for power
series in general, the first question is:
IsT(x)convergent atx=a? There is no guarantee except ata=x 0. The second question is:
IfT(x)converges atx=a, does it equalf(a)? The answer is negative as show by the function:
f 1 (x) =
e−x^12 x 6 = 0
0 x= 0