The series 1 + x^2 + x^4 + ....... sums to 1 / ( 1 - x^2)
The series -[x + x^3 + x^5 + ] .........sums to - [ x / (1 -x^2 )
So
x = 1 /(1 - x^2) - x / (1 -x^2)
x = (1 - x) / (1 -x^2)
x ( 1 - x^2) = (1 - x)
x - x^3 = 1 - x
x^3 - 2x + 1 = 0 (1)
x = 1 is a solution to (1)
Using synthetic division, we have
1 [ 1 0 - 2 1 ]
1 1 -1
____________
1 1 -1 0
The remaining polynomial is
x^2 + x - 1 = 0 complete the square on x
x^2 + x + 1/4 = 1 + 1/4
( x + 1/2)^2 = 5/4 take both roots
x + 1/2 = ±√5 /2
x = -1 ±√5
_____ ( 2)
2
But x = 1 makes the original equation undefined....so....the only solutions are represented by (2)