Let,

x = 0.231231231231…..

x = 0.231 + 0.000231 + 0.000000231 + …∞

⇒ x = 231(0.001 + 0.00001 + 0.0000001 + …∞ )

⇒ x =

We observe that the above progression possess a common ratio. So it is a geometric progression.

Common ratio = 1/1000 and first term (a) = 1/1000

Sum of infinite GP = ,where a is the first term and k is the common ratio.

Note: We can only use the above formula if |k|<1

∴ we can use the formula for the sum of infinite GP.

⇒ x = 231×

∴ x = 231/999