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.
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.