(In reply to

analytical proof .....a hint by Ady TZIDON)

I will first note that both P and Q converge absolutely. This means I can make arbitrary changes in the pattern of additions or subtractions without needing to worry that I accidentally changed the limit of the summation.

Lets see what Ady's hint "evaluate P-Q" does. I will take the difference P-Q as a series of differences of nth terms:

P-Q = (1 - 1) + (-2^-2 - 2^-2) + (4^-2 + 4^-2) + (-5^-2 + 5^-2) + (

7^-2 - 7^-2) + (-8^-2 - 8^-2) + (10^-2 + 10^-2) + (-11^-2 + 11^-2) + ....

P-Q = 0 + -2*(2^-2) + 2*(4^-2) + 0 + 0 + -2*(8^-2) + 2*(10^-2) + 0

P-Q = -(1/2)*1 + (1/2)*(2^-2) + -(1/2)*(4^-2) + (1/2)*(5^-2) + ...

P-Q = -(1/2)*[1 - 2^-2 + 4^-2 - 5^-2 + ...]

P-Q = -(1/2)*P

There it is! Easily solve for Q = (3/2)P. Then P/Q = P/((3/2)*P) = 2/3.