Question

How do you find the nth Taylor polynomials for f(x) = ln x centered about a=1?

Answer 1

`lnx = sum_(n=1)^oo (-1)^(n-1)(x-1)^n/n`

Explanation:

We have:

`f(x) = lnx`

`f'(x) = 1/x = x^-1`

`f''(x) = -x^-2`

and in general:

`f^((n)) = (-1)^(n-1)(n-1)!x^-n`

So that:

`f(1) = 0`

`f^((n)) (0) = (-1)^(n-1)(n-1)!`

The Taylor series is then:

`sum_(n=1)^oo (-1)^(n-1)((n-1)!)/(n!) (x-1)^n = sum_(n=1)^oo (-1)^(n-1)(x-1)^n/n`