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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![The expression presented in item 27 is a mathematical series. It is written as follows:
\[ \sum_{n=3}^{\infty} \left( \frac{3}{11} \right)^{-n} \]
This represents the sum of the terms of a geometric series starting from \( n = 3 \) and continuing infinitely.
In more detail:
- The index of summation \( n \) starts at 3 and goes to infinity (\(\infty\)).
- Each term in the series is of the form \(\left(\frac{3}{11}\right)^{-n} \), where the base is \(\frac{3}{11}\) and it is raised to the power of \(-n\).
This means that for each increment of \( n \), the term \(\left(\frac{3}{11}\right)^{-n} \) is added to the sum.
### Explanation:
- **Geometric Series:** A geometric series is a series with a constant ratio between successive terms.
- **Summation Notation:** The symbol \(\sum\) signifies summation, which means adding up a series of terms.
- **Power Notation:** The term \(\left(\frac{3}{11}\right)^{-n}\) indicates that \(\frac{3}{11}\) is raised to the power of \(-n\).
To further understand, consider the behavior of the term \(\left(\frac{3}{11}\right)^{-n} \):
- Raising a number less than 1 (3/11) to a negative power effectively inverts the fraction and raises it to a positive power (i.e., \(\left(\frac{3}{11}\right)^{-n} = \left(\frac{11}{3}\right)^{n}\)).
Hence, the series \(\sum_{n=3}^{\infty} \left(\frac{3}{11}\right)^{-n}\) can be rewritten and analyzed to find its sum if it converges.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37708386-d0d8-46c3-a2d8-8e337c754037%2F1220f0fb-b634-4fbd-b84c-026cdde1fb48%2Fjwku46_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The expression presented in item 27 is a mathematical series. It is written as follows:
\[ \sum_{n=3}^{\infty} \left( \frac{3}{11} \right)^{-n} \]
This represents the sum of the terms of a geometric series starting from \( n = 3 \) and continuing infinitely.
In more detail:
- The index of summation \( n \) starts at 3 and goes to infinity (\(\infty\)).
- Each term in the series is of the form \(\left(\frac{3}{11}\right)^{-n} \), where the base is \(\frac{3}{11}\) and it is raised to the power of \(-n\).
This means that for each increment of \( n \), the term \(\left(\frac{3}{11}\right)^{-n} \) is added to the sum.
### Explanation:
- **Geometric Series:** A geometric series is a series with a constant ratio between successive terms.
- **Summation Notation:** The symbol \(\sum\) signifies summation, which means adding up a series of terms.
- **Power Notation:** The term \(\left(\frac{3}{11}\right)^{-n}\) indicates that \(\frac{3}{11}\) is raised to the power of \(-n\).
To further understand, consider the behavior of the term \(\left(\frac{3}{11}\right)^{-n} \):
- Raising a number less than 1 (3/11) to a negative power effectively inverts the fraction and raises it to a positive power (i.e., \(\left(\frac{3}{11}\right)^{-n} = \left(\frac{11}{3}\right)^{n}\)).
Hence, the series \(\sum_{n=3}^{\infty} \left(\frac{3}{11}\right)^{-n}\) can be rewritten and analyzed to find its sum if it converges.
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