Not quite - Order refers to the degree of the Taylor Polynomial.

You want the quadratic p2(x) = x-x^2/2

R2(x) = f ''' (t) x^3/3! 0 < t < 1

f ''' = 2/(1+x^3) 0n [0,1] f ''' (x) < 2 anf f'''(x) > 1/4

R2(1) = f ''' (t)/3! < 2/6 =1/3

p2(1) = 1-1/2 =1/2

ln(2) < 1/2 + 1/3 = 5/6

f ''' (t) > 1/(4*3!) = 1/24

1/2 +1/24 < ln(2)

13/24 < ln(2)