Let ,x = 0.6666…
⇒ x = 0.6 + 0.06 + 0.006 + …
⇒ x = 6(0.1 + 0.01 + 0.001 + 0.0001 + …∞)
⇒ x = 6 \(\left(\frac{1}{10} + \frac{1}{100} + \frac{1}{1000} + \frac{1}{10000} + .... \infty\right)\)
This is an infinite geometric series.
Here, a = 1/10 and r = 1/10
\(\therefore\) X = 6 \(\times\) 1/9 = 6/9 = 2/3
0.\(\bar 6\) = \(\frac{2}{3}\)