(1)/(2),(1)/(4),(3)/(4),(4)/(4)

Simplify ((3(2)/(3))^(2)-(2 (1)/(2)))/((4(3)/(4))^(2)-(3(1)/(3))^(2))+(3(2)/(3)-2(1)/(2))/(4(3)/(4)-3(1)/(3))

(-(1)/(2)-(2)/(3)+(4)/(5)-(1)/(3)+(1)/(5)+(3)/(4))/((1)/(2)+(2)/(3)-(4)/(3)+(1)/(3)-(1)/(5)-(4)/(5)) is simplified to

3log 2 +(1)/(4) -(1)/(2)((1)/(4))^(2)+(1)/(3)((1)/(4))^(3)-….. =

3log 2 +(1)/(4) -(1)/(2)((1)/(4))^(2)+(1)/(3)((1)/(4))^(3)-….. =

Simplify : ((1)/(3)+(3)/(4)((2)/(5)-(1)/(3)))/(1(2)/(3) "of"(3)/(4)-(1)/(4)"of"(4)/(5))

[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))}]

Prove that, lim_(xtooo)((1)/(4)+(1)/(4^(2))+(1)/(4^(3))+.......+(1)/(4^(x)))=(1)/(3)

lim_(x rarr a){[(a^((1)/(2))+x^((1)/(2)))/(a^((1)/(4))-x^((1)/(4))))^(-1)-(2(ax)^((1)/(4)))/(x^((3)/(4))-a^((1)/(4))x^((1)/(2))+a^((1)/(2))x^((1)/(4))-a^((3)/(4)))]^(-1)-sqrt(2)^(log_(4)a)}^(8)

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