The terms in the series would be
[tex]81+81\left(\dfrac23\right)+81\left(\dfrac23\right)^2+81\left(\dfrac23\right)^3+\cdots[/tex]
Consider the first [tex]n[/tex] terms of this series, and call it
[tex]S_n=81+81\left(\dfrac23\right)+81\left(\dfrac23\right)^2+\cdots+81\left(\dfrac23\right)^{n-1}[/tex]
We have
[tex]\dfrac23S_n=81\left(\dfrac23\right)+81\left(\dfrac23\right)^2+81\left(\dfrac23\right)^3+\cdots+81\left(\dfrac23\right)^n[/tex]
and subtracting this from [tex]S_n[/tex] gives
[tex]S_n-\dfrac23S_n=81-81\left(\dfrac23\right)^n[/tex]
[tex]\dfrac13S_n=81\left(1-\left(\dfrac23\right)^n\right)[/tex]
[tex]S_n=243\left(1-\left(\dfrac23\right)^n\right)[/tex]
As [tex]n\to\infty[/tex], the [tex]\left(\dfrac23\right)^n[/tex] term will converge to 0 and leave you with
[tex]\displaystyle\lim_{n\to\infty}S_n=243[/tex]
