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##.