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

differentiating with respect to x

S'=2(x+x³+x⁵+...)=2x(1+x²+x⁴+...),,,,,converges |x|<1

same as

S'=2x/(1-x²)

integrating

S=-ln(1-x²)+C=ln(1/(1-x²))+C

x=0, S=0 from original S

C=0

S=ln(1/(1-x²))

if x=(√2)/2 for example

then S=ln(2)

Answer 2

If you differentiate the power series you get 2*sum(x^2n-1).

That is a geometric series the sum of which is 2x/(1-x) so that f'(x)=2x/(1-x)

The integral of that expression is 2[(1-x)-ln(1-x)]+C which is f(x).

Since f(0)=0, it is possible to find C.