# How do you find the Taylor series of #f(x)=ln(x)# ?

##### 1 Answer

Sep 10, 2014

Any Taylor series of a function

where

Let's say you need to approximate

- The Taylor series of degree 0 is simply
#f(1) = ln(1) = 0# - The Taylor series of degree 1 is the Taylor series of degree 0, plus
#(f'(1)(x-1)^1)/(1!)#

We know that#f'(x) = 1/x# for#x>0# , that#1! =1# , and that#(x-1)^1=(x-1)#

So the Taylor series of degree 1 is

#0+(1(x-1))/1 = (x-1)# . - The Taylor series of degree 2 is the Taylor series of degree 1, plus
#(f''(1)(x-1)^2)/(2!)#

We know that#f''(x) = -1/x^2# for#x>0# , and that#2! =2#

So the Taylor series of degree 2 is#(x-1)+(-1(x-1)^2)/2#

#=(x-1)-(x-1)^2/2# . - The Taylor series of degree 3 is the Taylor series of degree 2, plus
#(f'''(1)(x-1)^3)/(3!)#

We know that#f'''(x)=2/x^3# for#x>0# , and that#3! =6#

So the Taylor series of degree 3 is#(x-1)-(x-1)^2/2+(2/1(x-1)^3)/6#

#=(x-1)-(x-1)^2/2+(x-1)^3/3# .

Keep on working in this vein until you reach the degree that has been asked for. Usually during high school and early university, you won't need it beyond about the sixth degree as it gets quite time-consuming to calculate by hand.