What is the Taylor Series for f(x)=(x−1)^3 centered at x=0? What is the radius of convergence?

Kareem Mejia

Kareem Mejia

Answered question

2022-11-02

What is the Taylor Series for f ( x ) = ( x 1 ) 3 centered at x = 0? What is the radius of convergence?

Answer & Explanation

Raven Hawkins

Raven Hawkins

Beginner2022-11-03Added 19 answers

The taylor series can expressed as
n = 0 x n 6 ( 1 ) 3 n / ( ( 3 n ) ! n ! )
radius of convergence should be:
lim n   | ( x n + 1 6 ( 1 ) 2 n / ( ( 2 n ) ! ( n + 1 ) ! ) ) / ( x n 6 ( 1 ) 3 n / ( ( 3 n ) ! n ! ) ) | = lim n | x / ( n + 1 ) | = | x | < lim n n + 1
The radius of convergence should therefore be infinity!

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