Find a power series representation of f.

Question

Find a power series representation of f. Use a geometric series and properties of a power series.

(Assume that −1 ≤ x ≤ 1.)

f(x) = 8 ln(1 + x⁴)

Answer 1

y' = (32x^3)/(1+x^4)

y'' = (96x^2 - 42x^6)/(1+x^4)^2

y''' contains functions of x whose intercepts are zero; that is they do not contain final constant terms.

therefore the derivatives are zero at x=0.

Answer 2

What is the series for ln(1+x)? No replace x with x^4