If `1/1^2+1/2^2+1/3^2+...oo=pi^2/6` then value of `1-1/2^2+1/3^2-1/4^2+...oo=`

Explanation

(i) `(1)/(1^(2))+(1)/(2^(2)) +(1)/(3^(2) )+ ........+ infty =(pi^(2))/( 6)" "....(i) `
` therefore " "(1)/(1^(2)) +(1)/( 3 ^(2)) +(1)/( 5^(2)) +....+ infty `
` " " = ((1)/(1^(2))+ (1)/(2^(2))+(1)/(3^(2)) +(1)/(4^(2)) + ....+ infty ) -( (1)/(2^(2))+ ( 1) /(4^(2)) +(1)/( 6^(2)) + .......+ infty ) `
` =(pi^(2) )/(6) -(1)/(4) (pi^(2))/( 6) =(3)/(4) xx (pi^(2))/( 6) =(pi^(2))/( 8) `
` (ii) 1-( 1)/(2^(2)) +(1)/(3^(2) )-(1)/( 4^(2) )+ ....+ infty =(( 1)/(1^(2) ) -(1)/(3^(3) ) + .....+ -(1)/(2^(2)) -(1)/(2^(2)) + .....infty ) `
` " " =( pi^(2))/(8) -(1)/(4) xx ( pi ^(2))/( 6) =( pi ^(2))/( 12)`.