Find the value of `{:(" "SigmaSigma),(1 le i le j):} " "i xx (1/2)^j`

Answer

2

Explanation

`S=sumsum_(ileiltj)ixx(1/2)^(j)`
`=1xx[(1/2)^(2)+(1/2)^(3)+(1/2)^(4)+…]`
`+2xx[(1/2)^(3)+(1/2)^(4)+(1/2)^(5)+..]`
`+3xx[(1/2)^(4)+(1/2)^(5)+(1/2)^(6)+…]`
…..
`=1xx((1/2)^(2))/(1-(1/2))+2xx((1/2)^(3))/(1-(1/2))+3xx((1/2)^(4))/(1-(1/2))+..`
`thereforeS=1xx(1/2)+2xx(1/2)^(2)+3xx(1/2)^(3)+...`
`therefore (1/2)S=1xx(1/2)^(2)+2xx(1/2)^(3)+3xx(1/2)^(4)...`
Subtracting, we get
`(1/2)S=(1/2)+(1/2)^(2)+(1/2)^(3)+(1/2)^(4)+...=(1/2)/(1-1/2)=1`
`therefore` S=2