Question

Write the Taylor Series for f(x) = In (x)centered at a = 2. Show the first five terms.

Answer 1

`f(x) = ln(2) + (1)/(2) (x-2) - (1)/(8) (x-2)^2 + (1)/(24) (x-2)^3 - (1)/(64) (x-2)^4 + ...`

Explanation:

The Taylor Series for a function
`f(x)` centered at
`x=a` is:

`f(x) = \sum_(n=0)^(\infty) (f^((n))(a))/(n!) (x-a)^(n)`

Let's find the first four derivatives of the function:

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

`f''(x) = -(1)/(x^2)`

`f'''(x) = (2)/(x^3)`

`f^(iv)(x) = -(6)/(x^4)`

By replacing
`x=2` in the previous derivatives and then completing the expansion of the Taylor series of the function, we get:

`f(x) = f(2) + (f'(2))/(1!) (x-2) + (f''(2))/(2!) (x-2)^2 + (f'''(2))/(3!) (x-2)^3 + (f^(iv)(2))/(4!) (x-2)^4 + ...`

`f(x) = ln(2) + (1)/(2) (x-2) - (1)/(8) (x-2)^2 + (1)/(24) (x-2)^3 - (1)/(64) (x-2)^4 + ...`