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# Show that the series obtained from a generalization of the decimal expansion converges

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

# Describe a geometric connection with a generalization of the decimal expansion of a number

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

Describe a geometric method for obtaining .

Instead of dividing the real line into segments with 10 subintervals and taking the greatest integer number of intervals, we divide the line into subintervals.

# Prove properties of a generalization of the decimal expansion of a number

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 for each .

Proof. Suppose otherwise, that for some we have . Hence,

This contradicts that is the greatest integer such that

# Prove that decimals ending in zeros can also be written as a decimal ending in repeated nines

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.

Proof. Let

where for all . Then we have,

where is an integer. Then,

# Prove that every repeating decimal represents a rational number

Prove that every repeating decimal represents a rational number.

Proof. Let

be any repeating decimal. Let

Then,

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 the given decimal number as a quotient of two integers

Let

Express as an infinite series, find the sum, and express as a quotient of two integers.

We have