Evaluate `sum_(m=1)^(oo)sum_(n=1)^(oo)(m^(2)n)/(3^(m)(n*3^(m)+m*3^(n)))`.

Explanation

Let`S=sum_(m=1)^(oo)sum_(n=1)^(oo)(m^(2)n)/(3^(m)(n*3^(m)+m*3^(n)))`
`=sum_(m=1)^(oo)sum_(n=1)^(oo)(1)/(((3^(m))/(m))((3^(m))/(m)+(3^(n))/(n)))`
Now, let `a_(m)=(3^(m))/(m)" and "a_(n)=(3^(n))/(n)`
Then. `S=sum_(m=1)^(oo)sum_(n=1)^(oo)(1)/(a_(m)(a_(m)+a_(n))) " " "....(i)"`
By interchanging m and n, then
`S=sum_(m=1)^(oo)sum_(n=1)^(oo)(1)/(a_(n)(a_(n)+a_(m)))" " ".....(ii)"`
On adding Eqs. (i) and (ii), we get
`2S=sum_(m=1)^(oo)sum_(n=1)^(oo)(1)/(a_(m)a_(n))=sum_(m=1)^(oo)sum_(n=1)^(oo)(mn)/(3_(m)3_(n))`
`=(sum_(n=1)^(oo)(n)/(3^(n)))^(2)=[1((1)/(3))+2((1)/(3))^(2)+3((1)/(3))^(3)+"..."]^(2)`
`=(S')^(2)" " "......(iii)"`
where, `S'=1((1)/(3))+2((1)/(3))^(2)+3((1)/(3))^(3)+"...+"oo`
`((1)/(3ul)S'=1((1)/(3ul))^(2)+2((1)/(3ul))^(3)+" "+"...+"oo)/((2)/(3)S'=(1)/(3)+((1)/(3))^(2)+((1)/(3))^(3)+" "+"...+"oo)`
`=((1)/(3))/(1-(1)/(3))=(1)/(2)`
`:.S'=(3)/(4)`
From E q. (iii), we get `2s=((3)/(4))^(2)`
`S=(9)/(32)`