What is binary(base2) representation of this endless ternary (that is base3) number 0.111111111..1..?

Answer 1

0.111111... (base 3) is equivalent to:
1/3 + 1/9 + 1/27 + 1/81 + ...

This is equal to 1/2

In binary it would simply be:
0.1 (base 2)

Note: It's very interesting but if you have 0.111111... in base n, then it will be 0.1 in base n-1.

For example:
0.11111... (base 10) is 1/9, which is 0.1 in base 9
0.11111... (base 9) is 1/8, which is 0.1 in base 8
etc.