# Repeating decimal as infinite geometric series | Precalculus | Khan Academy | Summary and Q&A

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November 8, 2013
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Repeating decimal as infinite geometric series | Precalculus | Khan Academy

## TL;DR

The video explains how to represent a repeating decimal as an infinite geometric series and then expresses it as a fraction.

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### Q: How can a repeating decimal be represented as an infinite series?

A repeating decimal can be represented as an infinite series by considering the repeating pattern as individual terms in the series, with four zeroes before the decimal each time.

### Q: Is the infinite series representing a repeating decimal a geometric series?

Yes, the infinite series representing a repeating decimal is a geometric series because the common ratio between each term is consistent. In this case, the common ratio is 10 to the negative fourth power.

### Q: How can the sum of an infinite geometric series be expressed as a fraction?

The sum of an infinite geometric series can be expressed as a fraction using the formula a / (1 - r), where "a" is the first term and "r" is the common ratio. In the case of the repeating decimal 0.4008, the fraction is 4,008 / 9,999.

### Q: Can the fraction 4,008 / 9,999 be simplified?

Yes, the fraction 4,008 / 9,999 can be simplified by dividing both the numerator and denominator by 3. The simplified fraction is 1,336 / 3,333.

## Summary & Key Takeaways

• The video demonstrates how to represent a repeating decimal, such as 0.4008, as an infinite sum or series.

• It shows that the repeating pattern 4008 can be viewed as individual terms in the series, with four zeroes before the decimal each time.

• The video explains that this series is a geometric series with a common ratio of 10 to the negative fourth power.