SOLUTION: 1-(1/4)+(1/9)-(1/16)+(1/25)-............=?

Question 1061883: 1-(1/4)+(1/9)-(1/16)+(1/25)-............=?

Found 2 solutions by rothauserc, ikleyn:
Answer by rothauserc(4718)   (Show Source): You can put this solution on YOUR website!
1-(1/4)+(1/9)-(1/16)+(1/25)-............=?
:
we can rewrite this as
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summation i=1,...+infinity of (-1)^(i+1) / i^2
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The approach is to break the sum into even and odd parts
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we know that summation i=1,...+infinity of 1/i^2 = pi^2/6
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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
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Answer by ikleyn(52792)   (Show Source): You can put this solution on YOUR website!
.
1-(1/4)+(1/9)-(1/16)+(1/25)- . . . = ?
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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 = - .( = . = . = .
Answer.   .

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