Successive terms have a common factor of [tex]-\dfrac23[/tex], which means the [tex]n[/tex]th partial sum of the series can be written as
[tex]S_n=9-6+4-\dfrac83+\cdots+9\left(-\dfrac23\right)^{n-1}[/tex]
[tex]S_n=9\left(1+\left(-\dfrac23\right)^1+\left(-\dfrac23\right)^2+\left(-\dfrac23\right)^3+\cdots+\left(-\dfrac23\right)^{n-1}[/tex]
[tex]\implies-\dfrac23S_n=9\left(\left(-\dfrac23\right)^1+\left(-\dfrac23\right)^2+\left(-\dfrac23\right)^3+\left(-\dfrac23\right)^4+\cdots+\left(-\dfrac23\right)^n\right)[/tex]
[tex]\implies S_n-\left(-\dfrac23\right)S_n=9\left(1-\left(-\dfrac23\right)^n\right)[/tex]
[tex]\dfrac53S_n=9\left(1-\left(-\dfrac23\right)^n\right)[/tex]
[tex]S_n=\dfrac{27}5\left(1-\left(-\dfrac23\right)^n\right)[/tex]
As [tex]n\to\infty[/tex], the geometric term vanishes, leaving you with
[tex]S=\displaystyle\lim_{n\to\infty}S_n=\dfrac{27}5[/tex]
