Find the power series representation about the center c=0 of

Question

(a) f(x)= 1/(1+x^2)

(b) Use part (a) to find the power series of g(x)=ln(1+x^2)

-Thank You

Answer 1

taking various derivatives, it is clear that

1/(1+x^2) = 1 - x^2 + x^4 - x^8 + ... for |x|<1

now, since d/dx ln(1/(1+x^2)) = 2x/(1+x^2)

2x/(1+x^2) = 2x - 2x^3 + 2x^5 - 2x^7 + ...
and
ln(1/(1+x^2)) = ∫ 2x/(1+x^2) dx = x^2 - x^4/2 + x^6/3 - x^8/4 + ...