*Originally posted by FabianFnas*

**1/9, you say it is 0/2 + 0/4 + 0/8 + 1/16 + 1/32 + 0/64 + 1/128 + ... = 1/9
**

it is actually not so.

The correct is 0/2 + 0/4 + 0/8 + 1/16 + 1/32 + 1/64 + 0/128 + ... = 1/9

or if we show only the indices = 000 111 000 111 ...

Go to the binary system, 1/9 (10) = 1/1001 (2), do the division 1/1001 and you get the result .000111000111... (a pattern of 3 ...[text shortened]... arithemtics again.

64x-x = 63x = 7

x = 7/63 = 1/9

And voilà We've shown that 1/9 is realy

The pattern that he is asked for is different than the one you have shown, the series for the numerators according to his stipulations are

0,0,0,1,1,0,1,1,2,1,0,1,2,2,2,3,4,1,1,3,4,3,1,5,5,2,5,2 for the first 28

that doesnt look like 0,0,0,1,1,1,0,0,0,1,1,1..........