(i) Use the geometric series to express the number 1 /(10^6-1) as a decimal fraction. (ii) Show

step 1 For the first part of this question we need to use the geometric series to express a number whic

(i) Use the geometric series to express the number 1...

(i) Use the geometric series to express the number $1 /\left(10^{6}-1\right)$ as a decimal fraction.
(ii) Show that the decimal representation of $1 / 7$ can be written as $142857 /\left(10^{6}-1\right)$ (see Section 1.4).

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Key Concepts

Geometric Series

A geometric series is a sum of terms that have a constant ratio between successive terms. In the context of the problem, a geometric series is used to express a fraction like 1/(10^6 - 1) as an infinite sum, where each term corresponds to a power of 10^(-6). The formula for summing a geometric series, a/(1 - r), is key in rewriting the fraction in a form that reveals its decimal structure.

Repeating Decimal Representation

Repeating decimals occur when a rational number's decimal expansion consists of a finite sequence of digits that repeats indefinitely. The fact that denominators of the form 10^n - 1 produce repeating decimal expansions is significant; it allows us to represent numbers like 1/7 in a compact form by aligning the repeating block with an appropriate power of 10, as seen in the representation 142857/(10^6 - 1).

Conversion of Fractions to Decimal Form

Converting fractions to decimal form often involves recognizing patterns or series that produce the repeating cycle. This includes using the geometric series method to sum infinite series or employing algebraic techniques to obtain a repeating block from a fraction. Understanding this conversion is essential in linking the algebraic fraction representation with its corresponding decimal expansion.

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