Let S be the sum of the first n terms of the left side:
[tex]S = \dfrac23 + \left(\dfrac23\right)^2 + \left(\dfrac23\right)^3 + \cdots + \left(\dfrac23\right)^n[/tex]
Multiply both sides by 2/3 :
[tex]\dfrac23 S = \left(\dfrac23\right)^2 + \left(\dfrac23\right)^3 + \left(\dfrac23\right)^4 + \cdots + \left(\dfrac23\right)^{n+1}[/tex]
Subtract this from S :
[tex]S - \dfrac23 S = \dfrac23 - \left(\dfrac23\right)^{n+1}[/tex]
Solve for S :
[tex]\dfrac13 S = \dfrac23 - \left(\dfrac23\right)^{n+1}[/tex]
[tex]S = 2 - 3 \left(\dfrac23\right)^{n+1}[/tex]
As n gets larger and larger, S converges to the given sum, and the term (2/3)ⁿ⁺¹ converges to zero, which leaves us with
[tex]\displaystyle \lim_{n\to\infty} S = \boxed{x = 2}[/tex]
