The Taylor series for f(x) = ln x about x0=1 given in

Chapter 8, Problem 35E

(choose chapter or problem)

The Taylor series for \(f(x)=\ln x\) about \(x_{0}=1\) given in equation (13) can also be obtained as follows:
(a) Starting with the expansion \(1 /(1-s)=\Sigma_{n=0}^{\infty} s^{n}\) and observing that

$$\frac{1}{x}=\frac{1}{1+(x-1)}$$ ,

obtain the Taylor series for \(1 / x\) about \(x_{0}=1\).
(b) Since \(\ln x=\int_{1}^{x} 1 / t d t\), use the result of part (a) and termwise integration to obtain the Taylor series for \(f(x)=\ln x\) about \(x_{0}=1\)

Equation Transcription:

∫

Text Transcription:

f(x)=ln ⁡x

x_0=1

1/(1−s)= Sigma_n^infinity=0^s^n

1/x=1/1+(x-1)

1/x

x_0=1

 ln⁡ x= integral_1^x  1/tdt

f(x)=ln ⁡x

x_0=1