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 P/Q (Posted on 2012-10-12)
Evaluate P/Q, given:

P= 1-2-2+4-2-5-2+7 -2-8-2+10-2 -11-2+13-2-14-2+...

Q= 1+2-2-4-2-5-2+7 -2+8-2-10-2 -11-2+13-2+14-2+...

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 re: analytical proof .... a hint .... the proof Comment 5 of 5 |

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.

 Posted by Brian Smith on 2016-11-25 22:44:30

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