Use differentiability of power series to find the sum

Question

Use differentiability of power series to find the sum (n=1 and inf on sum sign) ∑ (x²ⁿ)/n within its radius of convergence.

Answer 1

S(x) = sign) ∑_n=1^∞ (x²ⁿ)/n; S'(x) = ∑_n=1^∞(2x^2n-1) = 2x/(1 - x); IxI < 1

S(x) = ∫2x/(1 - x)dx = ∫(2/(1 - x) - 2)dx = - 2lnI1 - xI - 2x + C, but S(0) = 0; S(0) = - 2ln1 - 0 = 0;

S(x) = - 2lnI1 - xI - 2x, - 1 < x < 1