`6 (2)/(3)+(1)/(2)+4 (1)/(3)-1 (1)/(4)=`

The sum of the infinite series (1)/(2) ((1)/(3) + (1)/(4)) - (1)/(4)((1)/(3^(2)) + (1)/(4^(2))) + (1)/(6) ((1)/(3^(3)) + (1)/(4^(3))) - ...is equal to

Simplify : 8 (1)/(2)-[3 (1)/(4)-:{1(1)/(4) - (1)/(2) ( 1 (1)/(2)-(1)/(3)-(1)/(6))}]

9(3)/(4)-:[2(1)/(6)+{4(1)/(3)-(1(1)/(2)+1(3)/(4))}]

3(1)/(4)+2(1)/(2)+6(1)/(6)=?

[4(1)/(5)-:{1(3)/(4)-(1)/(2)(3(1)/(2)-(1)/(4)-(1)/(6))}]

[[ Simplify: [3(1)/(4)-:{1(1)/(4)-(1)/(2)(2(1)/(2)-(1)/(4)-(1)/(6))}]