good enough for late time

$x =\large { 1 - 2^{-2} + 4^{-2} - 5^{-2} + 7^{-2} - 8^{-2} + \ldots }$ $y = \large {1+ 2^{-2} - 4^{-2} - 5^{-2} + 7^{-2} + 8^{-2} - \ldots }$ $y - x = 2.2^{-2} - 2.4^{-2} + 2.8^{-2} - 2.10^{-2} + 2.14^{-2} - 2.16^{-2} + \ldots$ $y- x = 2.2^{-2} \Bigg( 1 -2^{-2} + 4^{-2} - 5^{-2} + 7^{-2} - 8^{-2} + \ldots \Bigg) = \frac{1}{2} x$ $y- x = \frac {1}{2} x$ $y = \frac {3}{2} x$ $\text {so we get } \frac {x}{y} = \frac {2}{3} = .666$ And the problem is solved :)