# How do you find the third degree Taylor polynomial for f(x)= ln x, centered at a=2?

ln(2)+1/2(x-2)-1/8(x-2)^2+1/24(x-2)^3.

The general form of a Taylor expansion centered at a of an analytical function f is f(x)=sum_{n=0}^oof^((n))(a)/(n!)(x-a)^n. Here f^((n)) is the nth derivative of f.

The third degree Taylor polynomial is a polynomial consisting of the first four (n ranging from 0 to 3) terms of the full Taylor expansion.

Therefore this polynomial is f(a)+f'(a)(x-a)+(f''(a))/2(x-a)^2+(f'''(a))/6(x-a)^3.

f(x)=ln(x), therefore f'(x)=1/x, f''(x)=-1/x^2, f'''(x)=2/x^3. So the third degree Taylor polynomial is: ln(a)+1/a(x-a)-1/(2a^2)(x-a)^2+1/(3a^3)(x-a)^3.

Now we have a=2, so we have the polynomial: ln(2)+1/2(x-2)-1/8(x-2)^2+1/24(x-2)^3.