Determine the Taylor series about the point x_{0} for the given function.

Kye

Kye

Answered question

2021-09-23

Determine the Taylor series about the point x0 for the given function. Also determine the radius of convergence of the series. 11x,x0=2

Answer & Explanation

funblogC

funblogC

Skilled2021-09-24Added 91 answers

f(x)=11x
f(x)=1(1x)2
f(x)=2(1x)3
f(x)=6(1x)4
f(n)(x)=n!(1x)n+1
Therefore, using formula for Taylor series f(x)=n=0f(n)(x0)n!(xx0)n:
f(n)(2)(1)n+1n!
f(x)=1(x2)+(x2)2+(x2)3=n=0(1)n+1(x2)n
Apply ratio test:
limn|(x2)n+1(x2)n|=|x2|
Series converges when |x2|<1
Radius of convergence is ρ=1.
Results:
n=0(1)n+1(x2)n,ρ=1

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