You can put this solution on YOUR website! 1-(1/4)+(1/9)-(1/16)+(1/25)-............=?
:
we can rewrite this as
:
summation i=1,...+infinity of (-1)^(i+1) / i^2
:
The approach is to break the sum into even and odd parts
:
we know that summation i=1,...+infinity of 1/i^2 = pi^2/6
:
The even part is (1/2)^2 + (1/4)^2 plus (1/6)^2, etc. Factoring out (1/4) shows that this even part sum is one fourth of the total sum. So, the odd part is 3/4 of the sum. 3/4 - 1/4 is a half, so the series converges to pi^2 / 12
:
Well known fact (after Euler) is
S = 1 + (1/4) + (1/9) + (1/16) + (1/25) + (1/36) + . . . = (1)
The sum of the even terms is
E = (1/4) + (1/16) + (1/36) + . . . =
(1/4)*(1 + (1/4) + (1/9) + . . . ) = (2)
What the problem actually asks about is the difference S - 2E = - .( = . = . = .
