We may generalize the decimal expansion of a number by replacing the integer 10 with any integer . If , let denote the greatest integer greater than . Assuming the integers have been defined, let , denote the largest integer such that

Show that the series

converges and has sum .

*Proof.* Since we have

Since

we have established the convergence of

There is a mistake here: the leftmost sum should run from 1, not from zero. It can be shown following the same reasoning as on page 393