Find the value of `underset((inejnek))(sum_(i=0)^(oo)sum_(j=0)^(oo)sum_(k=0)^(oo))1/(3^(i)3^(j)3^(k))`.

Explanation

`underset((inejnek))(sum_(i=0)^(oo)sum_(j=0)^(oo)sum_(k=0)^(oo))1/(3^(i)3^(j)3^(k))`
=sum when I,j,k are independent
`-3xx`(sum when any two of I,j,k are equal)
`+2xx` (sum when i=j=k)
`=sum_(i=0)^(oo)sum_(j=0)^(oo)sum_(k=0)^(oo)1/(3^(i)3^(j)3^(k))-3sum_(i=0)^(oo)sum_(k=0)^(oo)1/(9^(i)3^(k))+2sum_(i=0)^(oo)1/(27^i)`
`=(sum_(i=0)^(oo)1/3^(i))^(3)-3(sum_(i=0)^(oo)1/3^(k))+2(sum_(i=0)^(oo)1/27^(i))`
`=(3/2)^(3)-3(9/8)(3/2)+2(27/26)`
`=81/208`