Find the Taylor Series of f(x) = ln(x) at a = 5?

Question

C0 =
C1 =
C2 =
C3 =
C4 =

Answer 1

f(x) = f(5)+(x-5)f´(5) +(x-5)^2/2! f´´5 +(x-5)^3/3! f´´´(5)++
ln x= ln5+(x-5)*1/5 -(x-5)^2/2! *(1/5^2) +(x-5)^3/3! (2/5^3)++
(-1)^(n+1)*(x-5)^n/n! *(n-1)/5^n++

Answer 2

1/x = 1/[5 + (x-5)] = (1/5)/[1+(x-5)/5] = (1/5)[1 - (x-5)/5 + ((x-5)/5)^2 - ((x-5)/5)^3 + ((x-5)/5)^4 - ...]
Integrating it from 5 to x will give you the answer.