Some Basic Questions on Decimals through Addresses

4. (c) Suppose we drop the 1, 1/2, 1/16, 1/128 terms of the r = 1/2 series. Summing the terms in pairs, we have

1/4 + 1/8 + 1/32 + 1/64 + ... = 3/8 + 3/64 + ... = 3/7

Changing binary digits 3, 4, 6, 7, 8, 9, 10, 11, ... moves a point by 3/7.

Changing binary digits 3, 4, 5, 7, 8, 9, 11, 12, 13, ... moves a point by 7/15, because

1/4 + 1/8 + 1/16 + 1/64 + 1/128 + 1/256 + ... = 7/16 + 7/256 + ... = 7/15

Changing binary digits 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, ... moves a point by 15/31 because

1/4 + 1/8 + 1/16 + 1/32 + 1/128 + 1/256 + 1/512 + 1/1024 + ... = 15/32 + 15/1024 + ... = 15/31

Changing binary digits 3, ..., n, n+2, ..., 2n-1, ... moves a point by (2n-1 - 1)/(2n-1).

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