Use the definition of Taylor series to find the Taylor

ArcactCatmedeq8

ArcactCatmedeq8

Answered question

2021-11-17

Use the definition of Taylor series to find the Taylor series, centered at c for the function. f(x)=lnx,c=1

Answer & Explanation

Sact1989

Sact1989

Beginner2021-11-18Added 10 answers

The Taylor series of lnx around c=1. We know from the definition of Taylor series of the function f
f(x)=f(x)x=c+f(x)x=c1!(xc)+f(x)x=c2!(xc)2+f(x)x=c3!(xc)3+
Now the values of differential functions at given point
f(1)=ln(1)=0, f(1)=1x|x=1=1
f(1)=1x2x=1=1, f(1)=2x3|{x=1}=2f(1)=6x4|x=1=6
fn(1)=(1)n+1(n1)!xnx=1=(1)n((n1)!)
Thus the Taylor series expression become
f(x)=0+(x1)12!(x1)2+2!13!(x1)33!14!+(1)n+1(n1)!1n!(x1)n
Or we can write as
f(x)=(1x)12(x1)2+13(x1)314(x1)4+(1)n+1(n1)!1n!(x1)n

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