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

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

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Questions & Answers

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.