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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calculusssss please solve question 6
![**Mathematical Concepts in Series and Convergence**
**Question 6:**
Given that the improper integral \( \int_{1}^{\infty} x^{-p} dx \) converges, then "p" should be:
- a. \( p < 1 \)
- b. \( p < -1 \)
- c. \( p > 1 \)
- d. \( p > -1 \)
**Question 7:**
One of the following sequences is a divergent sequence:
- a. \( \left( \frac{2^n}{n!} \right) \)
- b. \( (-1)^n \frac{3n^4 + 2n}{n^5 + 7} \)
- c. \( \{(-1)^n e^{-3n} \} \)
- d. \( \left( (-1)^n \frac{n^2 + 4n}{5n + n^2} \right) \)
**Question 8:**
The sum of \( \sum_{n=0}^{\infty} \frac{(-1)^n 2n}{n!} \) is:
- a. \( e^{-π^2} \)
- b. \( -e^{π^2} \)
- c. \( \cos π \)
- d. \( -\cos π \)
**Question 9:**
For the series \( \sum_{n=0}^{\infty} \frac{(-1)^n n^2 + 3}{n + 10} \) one of the statements is true:
- a. converges by Integral test
- b. converges by Alternating series test
- c. diverges by Divergence test
- d. diverges by root test
**Question 10:**
The interval of convergence of the Power series: \( \sum_{n=1}^{\infty} \frac{(-1)^n (x - 4)^n}{n} \) is:
- a. (3,5)
- b. [3,5)
- c. [3,5]
- d. (3,5]
**Question 11:**
The series \( \sum_{n=1}^{\infty} \left( \frac{1}{3} \right)^n \) has sum equal to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F651a6648-31bf-414e-9733-041ee5de2545%2F15a78fd9-ff02-436e-a2d7-a6edf6f88ad3%2Fl5566_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematical Concepts in Series and Convergence**
**Question 6:**
Given that the improper integral \( \int_{1}^{\infty} x^{-p} dx \) converges, then "p" should be:
- a. \( p < 1 \)
- b. \( p < -1 \)
- c. \( p > 1 \)
- d. \( p > -1 \)
**Question 7:**
One of the following sequences is a divergent sequence:
- a. \( \left( \frac{2^n}{n!} \right) \)
- b. \( (-1)^n \frac{3n^4 + 2n}{n^5 + 7} \)
- c. \( \{(-1)^n e^{-3n} \} \)
- d. \( \left( (-1)^n \frac{n^2 + 4n}{5n + n^2} \right) \)
**Question 8:**
The sum of \( \sum_{n=0}^{\infty} \frac{(-1)^n 2n}{n!} \) is:
- a. \( e^{-π^2} \)
- b. \( -e^{π^2} \)
- c. \( \cos π \)
- d. \( -\cos π \)
**Question 9:**
For the series \( \sum_{n=0}^{\infty} \frac{(-1)^n n^2 + 3}{n + 10} \) one of the statements is true:
- a. converges by Integral test
- b. converges by Alternating series test
- c. diverges by Divergence test
- d. diverges by root test
**Question 10:**
The interval of convergence of the Power series: \( \sum_{n=1}^{\infty} \frac{(-1)^n (x - 4)^n}{n} \) is:
- a. (3,5)
- b. [3,5)
- c. [3,5]
- d. (3,5]
**Question 11:**
The series \( \sum_{n=1}^{\infty} \left( \frac{1}{3} \right)^n \) has sum equal to
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