Question

Use the definition of Taylor series to find the Taylor series, centered at c for the function. f(x)=ln x, c=1

Answer 1

Answer:

The taylor's series for f(x) = ln x centered at c = 1 is:

$ln (x) = \sum\limits^{\infty}_{n = 1} {\frac{(-1)^{n+1}(x-1)^n}{n} }$

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), $f^{iv}(x)$)

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

$ln (x) = \sum\limits^{\infty}_{n = 1} {\frac{(-1)^{n+1}(x-1)^n}{n} }$