Answer:
The taylor's series for f(x) = ln x centered at c = 1 is:
![ln (x) = \sum\limits^{\infty}_{n = 1} {\frac{(-1)^{n+1}(x-1)^n}{n} }](https://tex.z-dn.net/?f=ln%20%28x%29%20%3D%20%5Csum%5Climits%5E%7B%5Cinfty%7D_%7Bn%20%3D%201%7D%20%7B%5Cfrac%7B%28-1%29%5E%7Bn%2B1%7D%28x-1%29%5En%7D%7Bn%7D%20%7D)
Step-by-step explanation:
The calculations are handwritten for clarity and easiness of expression.
However, the following steps were taken in arriving at the result:
1) Write the general formula for Taylor series expansion
2) Since the function is centered at c = 1, find f(1)
3) Get up to four derivatives of f(x) (i.e. f'(x), f''(x), f'''(x),
)
4) Find the values of these derivatives at x =1
5) Substitute all these values into the general Taylor series formula
6) The resulting equation is the Taylor series
![ln (x) = \sum\limits^{\infty}_{n = 1} {\frac{(-1)^{n+1}(x-1)^n}{n} }](https://tex.z-dn.net/?f=ln%20%28x%29%20%3D%20%5Csum%5Climits%5E%7B%5Cinfty%7D_%7Bn%20%3D%201%7D%20%7B%5Cfrac%7B%28-1%29%5E%7Bn%2B1%7D%28x-1%29%5En%7D%7Bn%7D%20%7D)