Question 1061883
.
1-(1/4)+(1/9)-(1/16)+(1/25)- . . .  = ?
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<pre>
Well known fact (after Euler) is 

S = 1 + (1/4) + (1/9) + (1/16) + (1/25) + (1/36) + . . .  = {{{pi^2/6}}}           (1)

The sum of the even terms is 


E =     (1/4) +         (1/16) +          (1/36) + . . . = 

        (1/4)*(1 +      (1/4) +          (1/9) + . . . ) = {{{(1/4)*(pi^2/6)}}}    (2)


What the problem actually asks about is the difference  S - 2E = {{{pi^2/6}}} - {{{(2/4)}}}.({{{pi^2/6)}}} = {{{(2/4)}}}.{{{(pi^2/6)}}} = {{{(1/2)}}}.{{{(pi^2/6)}}} = {{{pi^2/12}}}.
</pre><U>Answer</U>.  &nbsp;&nbsp;{{{pi^2/12}}}.