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
For the sequence attached, write the first three terms: a1, a2, a3 (there is no need to simplify these expressions). Afer writing these terms, decide wheter the sequence converges or diverges. Compute lim n->inf an or explain why the limit fails to exist.
![### Series Representation
Given the formula for the term \(a_n\) in a series:
\[ (c) \quad a_n = (-1)^n \cdot \frac{2^n}{n!} \]
- **\( (-1)^n \)**: This term alternates the sign of each term in the series. When \( n \) is even, \((-1)^n\) is positive; when \( n \) is odd, \((-1)^n\) is negative.
- **\( 2^n \)**: This represents 2 raised to the power of \( n \), exponentially increasing with each term.
- **\( n! \)**: This denotes the factorial of \( n \), which is the product of all positive integers up to \( n \).
This formula is used to generate each term of the series based on its position \( n \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F81e8ba16-e7f4-4cb8-bc35-ac7ab5472c5c%2F81e19d29-06ac-4fe6-9c09-b0143d485978%2F32xb4m_processed.png&w=3840&q=75)
Transcribed Image Text:### Series Representation
Given the formula for the term \(a_n\) in a series:
\[ (c) \quad a_n = (-1)^n \cdot \frac{2^n}{n!} \]
- **\( (-1)^n \)**: This term alternates the sign of each term in the series. When \( n \) is even, \((-1)^n\) is positive; when \( n \) is odd, \((-1)^n\) is negative.
- **\( 2^n \)**: This represents 2 raised to the power of \( n \), exponentially increasing with each term.
- **\( n! \)**: This denotes the factorial of \( n \), which is the product of all positive integers up to \( n \).
This formula is used to generate each term of the series based on its position \( n \).
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