Some Basic Questions on Decimals through Addresses

4. (b) Recalling 1 + r + r2 + ... = 1/(1 - r) for |r| < 1, we see the sum is 4/3 for r = 1/4. Consequently, 1/4 + (1/4)2 + ... = 1/3. That is, changing the second, fourth, sixth, ... binary digits moves a point 1/3.

For r = 1/8 the sum is 8/7, so changing the third, sixth, ninth, ... binary digits moves a point by 1/7.

For r = 1/16 the sum is 16/15, so changing the fourth, eighth, twelvth, ... binary digits moves a point by 1/15.

For r = 1/2m, the sum is 2m/(2m - 1), so changing every mth digit moves a point by 1/n for n = 2m - 1.

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