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
we have established the convergence of
If the decimal expansion of a number ends in zeros, prove that this number can also be written as a decimal which ends in nines if we decrease the last nonzero digit in the decimal expansion by one unit. Prove this statement using infinite series.
where for all . Then we have,
where is an integer. Then,
Prove that every repeating decimal represents a rational number.
be any repeating decimal. Let
But this is a rational number since is an integer (since both and are integers) and is a rational number (and the sum of an integer and a rational number is rational)
Express as an infinite series, find the sum, and express as a quotient of two integers.