Taylor Series for log &#x2061;<!-- ⁡ --> ( x ) Does anyone know a closed form exp

boloman0z

boloman0z

Answered question

2022-06-25

Taylor Series for log ( x )
Does anyone know a closed form expression for the Taylor series of the function f ( x ) = log ( x ) where log ( x ) denotes the natural logarithm function?

Answer & Explanation

Xzavier Shelton

Xzavier Shelton

Beginner2022-06-26Added 26 answers

log ( 1 x ) = x + x 2 2 + x 3 3 + ( | x | < 1 )
There is no expansion around x = 1 because the log is singular at 0.
Craig Mendoza

Craig Mendoza

Beginner2022-06-27Added 6 answers

For x R satisfying 0 < x < 2
f ( x ) = ln ( x ) = ( x 1 ) 1 2 ( x 1 ) 2 + 1 3 ( x 1 ) 3 1 4 ( x 1 ) 4 +
f ( x ) = n = 1 [ ( 1 ) n + 1 n ( x 1 ) n ]

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