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Determine the convergence or divergence of the series.

Determine the convergence or divergence of the series.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Determine the convergence or divergence of the series.

### Sum of a Geometric Series The image contains a mathematical expression representing the sum of an infinite geometric series. The formula provided can be transcribed and explained as follows: #### Formula The mathematical expression is: \[ \sum_{n=0}^{\infty} (1.075)^n \] #### Explanation - **Summation Symbol \(\sum\):** This symbol represents the sum of a sequence. - **Index of Summation \(n\):** The variable \(n\) is the index of summation, starting from \(n=0\). - **General Term \((1.075)^n\):** Each term in the series is formed by raising \(1.075\) to the power of \(n\). #### Geometric Series In this case, \( (1.075)^n \) indicates a geometric series where: - The first term \(a = 1\) (since \( (1.075)^0 = 1 \)). - The common ratio \(r = 1.075\). Since this series has a common ratio greater than 1, it diverges to infinity. #### Representation This geometric series can be summed up indefinitely starting from \(n = 0\) to infinity, given the common ratio \(r\). ### Convergence and Divergence - A geometric series converges if the absolute value of the common ratio is less than 1 (\(|r| < 1\)). - The series diverges if the absolute value of the common ratio is 1 or greater (\(|r| \geq 1\)). In our case: \[ r = 1.075 > 1 \] Thus, the series diverges, meaning the sum goes to infinity. #### Practical Implications Understanding the convergence or divergence of series is essential in various branches of mathematics and applied sciences, such as calculating compound interest, population growth models, and solving differential equations.
Transcribed Image Text:### Sum of a Geometric Series The image contains a mathematical expression representing the sum of an infinite geometric series. The formula provided can be transcribed and explained as follows: #### Formula The mathematical expression is: \[ \sum_{n=0}^{\infty} (1.075)^n \] #### Explanation - **Summation Symbol \(\sum\):** This symbol represents the sum of a sequence. - **Index of Summation \(n\):** The variable \(n\) is the index of summation, starting from \(n=0\). - **General Term \((1.075)^n\):** Each term in the series is formed by raising \(1.075\) to the power of \(n\). #### Geometric Series In this case, \( (1.075)^n \) indicates a geometric series where: - The first term \(a = 1\) (since \( (1.075)^0 = 1 \)). - The common ratio \(r = 1.075\). Since this series has a common ratio greater than 1, it diverges to infinity. #### Representation This geometric series can be summed up indefinitely starting from \(n = 0\) to infinity, given the common ratio \(r\). ### Convergence and Divergence - A geometric series converges if the absolute value of the common ratio is less than 1 (\(|r| < 1\)). - The series diverges if the absolute value of the common ratio is 1 or greater (\(|r| \geq 1\)). In our case: \[ r = 1.075 > 1 \] Thus, the series diverges, meaning the sum goes to infinity. #### Practical Implications Understanding the convergence or divergence of series is essential in various branches of mathematics and applied sciences, such as calculating compound interest, population growth models, and solving differential equations.
On this page, we are exploring a mathematical series involving summation notation. The provided expression is written as: \[ \sum_{n=0}^{\infty} \frac{7}{5^n} \] ### Explanation: - **Summation Notation**: The capital Greek letter sigma (∑) signifies a summation. It is used to denote the sum of a sequence of terms. - **Lower Limit (n=0)**: The variable \( n \) starts at 0. - **Upper Limit (∞)**: The variable \( n \) goes to infinity, indicating this is an infinite series. - **Term Inside Summation**: The term to be summed is \( \frac{7}{5^n} \). ### Series Explanation: This is a geometric series where each term is of the form \( \frac{7}{5^n} \). Geometric series have a common ratio between consecutive terms. Here, the common ratio is \( \frac{1}{5} \). ### Convergence: For an infinite geometric series to converge, the common ratio \( r \) must satisfy \( |r| < 1 \). In this case, \( r = \frac{1}{5} \), which is less than 1, so the series converges. ### Sum of the Infinite Series: The sum \( S \) of an infinite geometric series \( \sum_{n=0}^{\infty} ar^n \) can be calculated using the formula: \[ S = \frac{a}{1 - r} \] where \( a \) is the first term and \( r \) is the common ratio. For this series: - The first term \( a \) is \( 7 \). - The common ratio \( r \) is \( \frac{1}{5} \). Plugging in these values: \[ S = \frac{7}{1 - \frac{1}{5}} = \frac{7}{\frac{4}{5}} = 7 \times \frac{5}{4} = \frac{35}{4} = 8.75 \] ### Conclusion: The sum of the infinite series \( \sum_{n=0}^{\infty} \frac{7}{5^n} \) is \( 8.75 \). This is a key property of convergent geometric series and
Transcribed Image Text:On this page, we are exploring a mathematical series involving summation notation. The provided expression is written as: \[ \sum_{n=0}^{\infty} \frac{7}{5^n} \] ### Explanation: - **Summation Notation**: The capital Greek letter sigma (∑) signifies a summation. It is used to denote the sum of a sequence of terms. - **Lower Limit (n=0)**: The variable \( n \) starts at 0. - **Upper Limit (∞)**: The variable \( n \) goes to infinity, indicating this is an infinite series. - **Term Inside Summation**: The term to be summed is \( \frac{7}{5^n} \). ### Series Explanation: This is a geometric series where each term is of the form \( \frac{7}{5^n} \). Geometric series have a common ratio between consecutive terms. Here, the common ratio is \( \frac{1}{5} \). ### Convergence: For an infinite geometric series to converge, the common ratio \( r \) must satisfy \( |r| < 1 \). In this case, \( r = \frac{1}{5} \), which is less than 1, so the series converges. ### Sum of the Infinite Series: The sum \( S \) of an infinite geometric series \( \sum_{n=0}^{\infty} ar^n \) can be calculated using the formula: \[ S = \frac{a}{1 - r} \] where \( a \) is the first term and \( r \) is the common ratio. For this series: - The first term \( a \) is \( 7 \). - The common ratio \( r \) is \( \frac{1}{5} \). Plugging in these values: \[ S = \frac{7}{1 - \frac{1}{5}} = \frac{7}{\frac{4}{5}} = 7 \times \frac{5}{4} = \frac{35}{4} = 8.75 \] ### Conclusion: The sum of the infinite series \( \sum_{n=0}^{\infty} \frac{7}{5^n} \) is \( 8.75 \). This is a key property of convergent geometric series and
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