The real number 0.9 can be written as an infinite series as shown. 6. 0.9 10 6. 6. 100 1000 Write the infinite series in sigma notation and use the formula for the sum of a geometric series to find the sum.
The real number 0.9 can be written as an infinite series as shown. 6. 0.9 10 6. 6. 100 1000 Write the infinite series in sigma notation and use the formula for the sum of a geometric series to find the sum.
Transcribed Image Text:**Infinite Series Representation of 0.9**
The real number 0.9 can be written as an infinite series as shown:
\[ 0.9 = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots \]
**Instructions**
Write the infinite series in sigma notation and use the formula for the sum of a geometric series to find the sum.
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**Explanation:**
The image above demonstrates how the number 0.9 can be expressed as an infinite series. Each term in the series is a fraction where the numerator is 9, and the denominator is a power of 10. This can be generalised in sigma notation.
In sigma notation, the series can be written as:
\[ 0.9 = \sum_{n=1}^{\infty} \frac{9}{10^n} \]
This series is a geometric series where the first term \(a = \frac{9}{10}\) and the common ratio \(r = \frac{1}{10}\).
The sum \(S\) of an infinite geometric series is given by the formula:
\[ S = \frac{a}{1 - r} \]
Plugging in the values for \(a\) and \(r\):
\[ S = \frac{\frac{9}{10}}{1 - \frac{1}{10}} = \frac{\frac{9}{10}}{\frac{9}{10}} = 1 \]
Thus, the sum of the infinite series is 1.
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![**Infinite Series Representation of 0.9**
The real number 0.9 can be written as an infinite series as shown:
\[ 0.9 = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots \]
**Instructions**
Write the infinite series in sigma notation and use the formula for the sum of a geometric series to find the sum.
---
**Explanation:**
The image above demonstrates how the number 0.9 can be expressed as an infinite series. Each term in the series is a fraction where the numerator is 9, and the denominator is a power of 10. This can be generalised in sigma notation.
In sigma notation, the series can be written as:
\[ 0.9 = \sum_{n=1}^{\infty} \frac{9}{10^n} \]
This series is a geometric series where the first term \(a = \frac{9}{10}\) and the common ratio \(r = \frac{1}{10}\).
The sum \(S\) of an infinite geometric series is given by the formula:
\[ S = \frac{a}{1 - r} \]
Plugging in the values for \(a\) and \(r\):
\[ S = \frac{\frac{9}{10}}{1 - \frac{1}{10}} = \frac{\frac{9}{10}}{\frac{9}{10}} = 1 \]
Thus, the sum of the infinite series is 1.](https://content.bartleby.com/qna-images/question/a4f75436-93ee-4e58-a0e3-afe0fa0e1ea5/4ac2ba8e-ced0-40cf-8d15-08fa78c08a86/6xcdo7d_processed.jpeg)
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