Question

I have these equations,

Equation 1:
F(x^k)/F'(x^k) = e^k - ((1/(2!))(F''(x^k))((e^k)^2))/(F'(x^k)) + ((1/(3!))(F'''(x^k))((e^k)^3))/(F'(x^k)) - ((1/(4!))(F''''(x^k))((e^k)^4))/(F'(x^k)) + O(||e^k||^5)

Equation 2:
(y^k) - (x^k) = -[e^k - ((1/(2!))(F''(x^k))((e^k)^2))/(F'(x^k)) + ((1/(3!))(F'''(x^k))((e^k)^3))/(F'(x^k)) - ((1/(4!))(F''''(x^k))((e^k)^4))/(F'(x^k)) + O(||e^k||^5)]

Using equations 1 and 2, how will I be able to get these equations?

((y^k) - (x^k))^2 = ((e^k)^2) - ((F''(x^k))((e^k)^3)/(F'(x^k))) + [(1/4)((F''(x^k))/(F'(x^k)))^2) + (((1/3)(F'''(x^k)))/(F'(x^k)))](e^k)^4 + O(||e^k||^5)

((y^k) - (x^k))^3 = ((e^k)^3) + (((3/2)(F''(x^k))((e^k)^4))/(F'(x^k))) + O(||e^k||^5)

((y^k) - (x^k))^4 = ((e^k)^4) + O(||e^k||^5)

I tried many times but failed. Thank you.

Answer 1

First off, dump that F' in the denominators. It appears on both sides of the equation, so it serves no purpose.

The rest is simply an application of

(a+b+c)^2 = a^2 + 2ab + 2ac + b^2 + 2bc + c^2
and so on

with some rearranging of terms.