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 displays a mathematical expression which represents an infinite series. The series is written as:
\[
\sum_{k=1}^{\infty} (-1)^{k+1} \frac{2k + 1}{k!}
\]
**Explanation:**
- \( \sum \) denotes the summation symbol, indicating that we will sum the terms from \( k = 1 \) to infinity.
- \( (-1)^{k+1} \) introduces an alternating sign for each term in the series, ensuring that the sign changes between positive and negative as \( k \) increases.
- \( \frac{2k + 1}{k!} \) is the main term in the series for each value of \( k \):
- \( 2k + 1 \) is a linear function of \( k \).
- \( k! \) (k factorial) represents the product of all positive integers less than or equal to \( k \), providing the denominator.
This series is a type of alternating series that might be encountered in calculus or higher-level mathematics courses, often in the context of studying convergence or series expansions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc711b872-7794-476e-be64-a71432f7213f%2Fe91c2d62-f313-421e-a733-f73bf1c1d437%2Fi8fr5t_processed.png&w=3840&q=75)
Transcribed Image Text:The image displays a mathematical expression which represents an infinite series. The series is written as:
\[
\sum_{k=1}^{\infty} (-1)^{k+1} \frac{2k + 1}{k!}
\]
**Explanation:**
- \( \sum \) denotes the summation symbol, indicating that we will sum the terms from \( k = 1 \) to infinity.
- \( (-1)^{k+1} \) introduces an alternating sign for each term in the series, ensuring that the sign changes between positive and negative as \( k \) increases.
- \( \frac{2k + 1}{k!} \) is the main term in the series for each value of \( k \):
- \( 2k + 1 \) is a linear function of \( k \).
- \( k! \) (k factorial) represents the product of all positive integers less than or equal to \( k \), providing the denominator.
This series is a type of alternating series that might be encountered in calculus or higher-level mathematics courses, often in the context of studying convergence or series expansions.
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