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(7) En=1 (n+1) +oo

(7) En=1 (n+1) +oo

Calculus: Early Transcendentals
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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Hi, I'm stuck on question #7 from the attached image.

**Problem 16.6: Study the convergence of the following series:** 1. \(\sum_{n=1}^{+\infty} (-1)^n n^5\) 2. \(\sum_{n=1}^{+\infty} \frac{1}{n^3}\) 3. \(\sum_{n=1}^{+\infty} \frac{10^n}{5^n}\) 4. \(\sum_{n=1}^{+\infty} \frac{2n+2}{10^n}\) 5. \(\sum_{n=1}^{+\infty} \frac{1}{6^n}\) 6. \(\sum_{n=1}^{+\infty} \sin^3\left(\frac{n^2}{n^3+1.4}\right)\) 7. \(\sum_{n=1}^{+\infty} \frac{n!}{(n+1)!}\) 8. \(\sum_{n=1}^{+\infty} \frac{n^2}{(n+1)!}\) 9. \(\sum_{n=1}^{+\infty} \sin^7\left(\frac{n^{4.2} + \sin(n^n) - 1}{n^6 + \cos(n^n) + 2}\right)\) 10. \(\sum_{n=1}^{+\infty} \frac{(-1)^n \sin n \cos(n)}{4^n + 5^n}\) 11. \(\sum_{n=1}^{+\infty} \frac{(\sqrt{7})^n}{(2n+3)!}\) 12. \(\sum_{n=1}^{+\infty} \frac{1}{(6n+1)!}\) This text is designed to introduce a collection of series for evaluation of convergence properties, ranging from simple power series, trigonometric functions, factorial expressions, and alternating series. Each item presents a unique challenge in determining whether the series converges or diverges, often requiring different techniques such as ratio tests, alternating series tests, or bounding comparisons.
Transcribed Image Text:**Problem 16.6: Study the convergence of the following series:** 1. \(\sum_{n=1}^{+\infty} (-1)^n n^5\) 2. \(\sum_{n=1}^{+\infty} \frac{1}{n^3}\) 3. \(\sum_{n=1}^{+\infty} \frac{10^n}{5^n}\) 4. \(\sum_{n=1}^{+\infty} \frac{2n+2}{10^n}\) 5. \(\sum_{n=1}^{+\infty} \frac{1}{6^n}\) 6. \(\sum_{n=1}^{+\infty} \sin^3\left(\frac{n^2}{n^3+1.4}\right)\) 7. \(\sum_{n=1}^{+\infty} \frac{n!}{(n+1)!}\) 8. \(\sum_{n=1}^{+\infty} \frac{n^2}{(n+1)!}\) 9. \(\sum_{n=1}^{+\infty} \sin^7\left(\frac{n^{4.2} + \sin(n^n) - 1}{n^6 + \cos(n^n) + 2}\right)\) 10. \(\sum_{n=1}^{+\infty} \frac{(-1)^n \sin n \cos(n)}{4^n + 5^n}\) 11. \(\sum_{n=1}^{+\infty} \frac{(\sqrt{7})^n}{(2n+3)!}\) 12. \(\sum_{n=1}^{+\infty} \frac{1}{(6n+1)!}\) This text is designed to introduce a collection of series for evaluation of convergence properties, ranging from simple power series, trigonometric functions, factorial expressions, and alternating series. Each item presents a unique challenge in determining whether the series converges or diverges, often requiring different techniques such as ratio tests, alternating series tests, or bounding comparisons.
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