Is the first term supposed to be 1/4, or did you actually mean 1/3?
I'll assume the latter. Write
[tex]S=\dfrac13+\dfrac19+\dfrac1{27}+\dfrac1{81}[/tex]
[tex]S=\dfrac13+\dfrac1{3^2}+\dfrac1{3^3}+\dfrac1{3^4}[/tex]
[tex]\dfrac13S=\dfrac1{3^2}+\dfrac1{3^3}+\dfrac1{3^4}+\dfrac1{3^5}[/tex]
[tex]\implies S-\dfrac13S=\dfrac13-\dfrac1{3^5}[/tex]
[tex]\dfrac23S=\dfrac13\left(1-\dfrac1{3^4}\right)[/tex]
[tex]S=\dfrac12\left(1-\dfrac1{81}\right)[/tex]
[tex]S=\dfrac{40}{81}[/tex]
