Question

What are the first three terms for the Taylor series for f(x)=lnx expanded about a=1 ?

Answer 1

`T(n) = x -1- (x- 1)^2/2 + (x- 1)^3/3`

Explanation:

Recall that the Taylor Series is given by

`T(n) = f(a) + f'(a)/(1!)(x- a) + f''(a)/(2!)(x- a)^2 + ... + f^n(a)/(n!) (x- a)^n`

Here `a =1`, so

`T(n) = ln(1) + (f'(1))/(1!)(x -1) + (f''(1))/(2!) (x - 2)^2 + (f'''(1))/(3!) (x- 2)^3`

We only need the first two terms, but since the first term is `0` we will have to calculate to the third derivative.

`f(x) = lnx -> f(1) = 0`
`f'(x) = 1/x -> f'(1) = 1`
`f''(x) = -1/x^2 -> f''(1) = -1`
`f'''(x) = 2/x^3 -> f'''(1) = 2`

Therefore:

`T(n) = 0 + (x - 1) - (x- 1)^2/2 + (x- 1)^3/3`

`T(n) = x -1- (x- 1)^2/2 + (x- 1)^3/3`

Hopefully this helps!