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

118.6K views
November 8, 2013
by
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.

## Key Insights

• 🔁 Repeating decimals can be represented as infinite series, with the repeating pattern considered as individual terms.
• 🥳 The series representing a repeating decimal can be a geometric series if the common ratio between terms is consistent.
• 😑 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.
• 😑 The repeating decimal 0.4008 can be expressed as the fraction 1,336 / 3,333.
• 🗂️ The fraction 1,336 / 3,333 can be simplified by dividing both the numerator and denominator by 3.
• ❓ The simplified fraction is 4,008 / 9,999.
• 🍉 The sum of the terms in the series representing 0.4008 is equivalent to the fraction 4,008 / 9,999.

## Transcript

Let's say we have the repeating decimal 0.4008, where the digits 4008 keep on repeating. So if we were to write it out, it would look something like this. 0.400840084008, and it keeps on going forever. What I want you to do right now is pause the video and think about whether you can represent this repeating decimal as an infinite sum, as an infini... Read More

### 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.