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 series converges, and if so find its sum.
![The image contains the mathematical expression for an infinite series:
\[
\sum_{k=1}^{\infty} \frac{4^{k+4}}{7^{k-1}}
\]
Explanation:
- The summation symbol \(\sum\) indicates that we are dealing with a series, which involves adding up terms.
- The index of summation, denoted by \(k\), starts at 1 and continues to infinity.
- The expression \(\frac{4^{k+4}}{7^{k-1}}\) is the general term of the series. For each integer value of \(k\), the expression inside determines the value of each term.
- In the numerator, \(4^{k+4}\) represents 4 raised to the power of \(k+4\).
- In the denominator, \(7^{k-1}\) represents 7 raised to the power of \(k-1\).
This infinite series adds up all terms of the form \(\frac{4^{k+4}}{7^{k-1}}\) as \(k\) starts from 1 and progresses to infinity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf9d26f1-3443-40e3-9fe1-74d82aba8491%2F298c6997-0019-4007-bb6e-617a9e461d47%2Fc1dqqjo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains the mathematical expression for an infinite series:
\[
\sum_{k=1}^{\infty} \frac{4^{k+4}}{7^{k-1}}
\]
Explanation:
- The summation symbol \(\sum\) indicates that we are dealing with a series, which involves adding up terms.
- The index of summation, denoted by \(k\), starts at 1 and continues to infinity.
- The expression \(\frac{4^{k+4}}{7^{k-1}}\) is the general term of the series. For each integer value of \(k\), the expression inside determines the value of each term.
- In the numerator, \(4^{k+4}\) represents 4 raised to the power of \(k+4\).
- In the denominator, \(7^{k-1}\) represents 7 raised to the power of \(k-1\).
This infinite series adds up all terms of the form \(\frac{4^{k+4}}{7^{k-1}}\) as \(k\) starts from 1 and progresses to infinity.
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