CK12 Calculus - Single Variable

Then.
ExerciseFind the Taylorseriesof the followingfunctionsat the givenx 0 with givendegreen.

1.f(x) =exatx= 0,n= 3
2.f(x) =ln xatx= 1,n= 4

3.f(x) = 1 +x+x^2 +x^3 +x^4 atx= -1,n= 4
Taylorand MaclaurinSeries
Definition(TaylorSeriesoff)
The Taylorseriesof a functionfatx=x 0 is the powerseries

takingall the termsof the Taylorpolynomials.The MaclaurinseriesM(x) offis the Taylorseriesatx= 0.
Example 1 Find the Maclaurinseriesoff(x) =cos x.

Solution..
Notice the pattern repeats every 4 terms. So
.
The Maclaurinseriesoff(x) =cos xis

.
ExerciseFind the Taylorseriesof followingfunctionsat the givenx 0.

1. at


2. at


3. at

Convergenceof Taylorand MaclaurinSeries

Sincef(n)(x) is definedfor all functionsfin this text, the TaylorseriesT(x) offis alwaysdefined.As for power
seriesin general,the first questionis: