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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![### Inequalities and Series Convergence
#### 6. Determine Divergence Using Inequalities
Which of the following inequalities can be used to show that the series \(\sum_{n=1}^{\infty} \frac{n}{n^2 + 1}\) diverges?
(a) \[
\frac{n}{n^2 + 1} \geq \frac{1}{n^2 + 1}
\]
(b) \[
\frac{n}{n^2 + 1} \leq \frac{1}{n}
\]
(c) \[
\frac{n}{n^2 + 1} \geq \frac{1}{2n}
\]
#### 7-40. Convergence or Divergence of Series
Determine whether each series converges or diverges.
7. \[
\sum_{n=1}^{\infty} \frac{1}{n^3 + 8}
\]
8. \[
\sum_{n=2}^{\infty} \frac{1}{\sqrt{n} - 1}
\]
9. \[
\sum_{n=1}^{\infty} \frac{n + 1}{n \sqrt{n}}
\]
10. \[
\sum_{n=1}^{\infty} \frac{n - 1}{n^3 + 1}
\]
11. \[
\sum_{n=1}^{\infty} \frac{9^n}{3^n + 10^n}
\]
12. \[
\sum_{n=1}^{\infty} \frac{6^n}{5^n - 1}
\]
13. \[
\sum_{n=2}^{\infty} \frac{1}{n \ln n}
\]
14. \[
\sum_{k=1}^{\infty} \frac{k \sin^2 k}{1 + k^3}
\]
15. \[
\sum_{k=1}^{\infty} \frac{\sqrt{k}}{\sqrt{k^3 + 4k + 3}}
\]
16. \[
\sum_{k=1}^{\infty} \frac{(2k - 1)(k - 1)}{(k + 1)(k^2 + 4)}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7534f13a-064e-4169-8c87-38e5472c444f%2F2d7fb559-ee35-490f-b0ae-868bc72d3fa1%2Fd5wkq4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Inequalities and Series Convergence
#### 6. Determine Divergence Using Inequalities
Which of the following inequalities can be used to show that the series \(\sum_{n=1}^{\infty} \frac{n}{n^2 + 1}\) diverges?
(a) \[
\frac{n}{n^2 + 1} \geq \frac{1}{n^2 + 1}
\]
(b) \[
\frac{n}{n^2 + 1} \leq \frac{1}{n}
\]
(c) \[
\frac{n}{n^2 + 1} \geq \frac{1}{2n}
\]
#### 7-40. Convergence or Divergence of Series
Determine whether each series converges or diverges.
7. \[
\sum_{n=1}^{\infty} \frac{1}{n^3 + 8}
\]
8. \[
\sum_{n=2}^{\infty} \frac{1}{\sqrt{n} - 1}
\]
9. \[
\sum_{n=1}^{\infty} \frac{n + 1}{n \sqrt{n}}
\]
10. \[
\sum_{n=1}^{\infty} \frac{n - 1}{n^3 + 1}
\]
11. \[
\sum_{n=1}^{\infty} \frac{9^n}{3^n + 10^n}
\]
12. \[
\sum_{n=1}^{\infty} \frac{6^n}{5^n - 1}
\]
13. \[
\sum_{n=2}^{\infty} \frac{1}{n \ln n}
\]
14. \[
\sum_{k=1}^{\infty} \frac{k \sin^2 k}{1 + k^3}
\]
15. \[
\sum_{k=1}^{\infty} \frac{\sqrt{k}}{\sqrt{k^3 + 4k + 3}}
\]
16. \[
\sum_{k=1}^{\infty} \frac{(2k - 1)(k - 1)}{(k + 1)(k^2 + 4)}
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