Recognizing the pattern is all you got to do

Although at first, the series down't show anything, yet if you change the third term in another form, you can clearly see that it is an AGP .

So, let $S$ denote the sum:

$S=\frac { 2 }{ 3 } -\frac { 5 }{ 6 } +\frac { 2 }{ 3 } -\frac { 11 }{ 24 } +...$

But, we can write the third term, and hence $S$ as:

$S=\frac { 2 }{ 3 } -\frac { 5 }{ 6 } +\frac { 8 }{ 12 } -\frac { 11 }{ 24 } +... \quad --------1.$

Dividing both sides with $2$ , we get:

$\frac { S }{ 2 } =\frac { 2 }{ 6 } -\frac { 5 }{ 12 } +\frac { 8 }{ 24 } -\frac { 11 }{ 48 } +... \quad --------2.$

Now, adding both the equations and noticing the formation of a $GP$ on the RHS, we get:

$\frac { 3S }{ 2 } =\frac { 2 }{ 3 } -(\frac { 1 }{ 6 } -\frac { 1 }{ 12 } +\frac { 1 }{ 24 } +...)$

The sum of the infinite $GP$ can be calculated, and on solving the equation, we get:

$S=\frac { 2 }{ 9 } =\frac { 2 }{ { 3 }^{ 2 } }$