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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Question
Determine whether the following series converges. Justify your answer.
![The image presents a mathematical expression involving an infinite series. It is labeled as item number 51.
The expression is:
\[ \sum_{k=1}^{\infty} \frac{7^k + 11^k}{11^k} \]
Explanation of components:
1. **Sigma Notation (\(\sum\))**: Represents the summation of a series.
2. **Limits of Summation (\(k=1\) to \(\infty\))**: Indicates that summation starts from \(k=1\) and continues indefinitely.
3. **Fraction (\(\frac{7^k + 11^k}{11^k}\))**: The term inside the summation is a fraction. In the numerator, there are two terms: \(7^k\) and \(11^k\). In the denominator, there is \(11^k\).
The series represents the sum of terms obtained by substituting successive integer values of \(k\) starting from 1 into the given expression, extending towards infinity. The primary focus is on understanding the convergence or evaluation of such series in mathematical analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc711b872-7794-476e-be64-a71432f7213f%2F7a485fae-3174-4ae5-9864-5d75165e59cc%2Fospxi6u_processed.png&w=3840&q=75)
Transcribed Image Text:The image presents a mathematical expression involving an infinite series. It is labeled as item number 51.
The expression is:
\[ \sum_{k=1}^{\infty} \frac{7^k + 11^k}{11^k} \]
Explanation of components:
1. **Sigma Notation (\(\sum\))**: Represents the summation of a series.
2. **Limits of Summation (\(k=1\) to \(\infty\))**: Indicates that summation starts from \(k=1\) and continues indefinitely.
3. **Fraction (\(\frac{7^k + 11^k}{11^k}\))**: The term inside the summation is a fraction. In the numerator, there are two terms: \(7^k\) and \(11^k\). In the denominator, there is \(11^k\).
The series represents the sum of terms obtained by substituting successive integer values of \(k\) starting from 1 into the given expression, extending towards infinity. The primary focus is on understanding the convergence or evaluation of such series in mathematical analysis.
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