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3-8 Test the series for convergence or divergence. dhe tadt Hour 3.- + - + 1

3-8 Test the series for convergence or divergence. dhe tadt Hour 3.- + - + 1

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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#3 please
im confused on when to use what tests

### Section 8.4: Exercises **1. (a)** What is an alternating series? (b) Under what conditions does an alternating series converge? (c) If these conditions are satisfied, what can you say about the remainder after n terms? **2.** What can you say about the series \( \sum a_n \) in each of the following cases? (a) \( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = 8 \) (b) \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 0.8 \) (c) \( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = 1 \) **3-8.** Test the series for convergence or divergence. 3. \( 4 - \frac{4}{8} + \frac{4}{9} - \frac{4}{10} + \frac{4}{11} - \cdots \) 4. \( 4 + \frac{5}{6} + \frac{7}{9} - \frac{11}{8} + \cdots \) 5. \( \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{2n+1} \) 6. \( \sum_{n=1}^{\infty} (-1)^n \frac{n}{\sqrt{n^3 + 2}} \) 7. \( \sum_{n=1}^{\infty} \frac{(-1)^n (3n - 1)}{2n + 1} \) 8. \( \sum_{n=1}^{\infty} (-1)^n \cos\left(\frac{\pi}{n}\right) \) **9-12.** Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? --- These exercises focus on alternating series, convergence tests, and series evaluation. They require an understanding of limit behaviors, ratio tests, and conditions for convergence.
Transcribed Image Text:### Section 8.4: Exercises **1. (a)** What is an alternating series? (b) Under what conditions does an alternating series converge? (c) If these conditions are satisfied, what can you say about the remainder after n terms? **2.** What can you say about the series \( \sum a_n \) in each of the following cases? (a) \( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = 8 \) (b) \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 0.8 \) (c) \( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = 1 \) **3-8.** Test the series for convergence or divergence. 3. \( 4 - \frac{4}{8} + \frac{4}{9} - \frac{4}{10} + \frac{4}{11} - \cdots \) 4. \( 4 + \frac{5}{6} + \frac{7}{9} - \frac{11}{8} + \cdots \) 5. \( \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{2n+1} \) 6. \( \sum_{n=1}^{\infty} (-1)^n \frac{n}{\sqrt{n^3 + 2}} \) 7. \( \sum_{n=1}^{\infty} \frac{(-1)^n (3n - 1)}{2n + 1} \) 8. \( \sum_{n=1}^{\infty} (-1)^n \cos\left(\frac{\pi}{n}\right) \) **9-12.** Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? --- These exercises focus on alternating series, convergence tests, and series evaluation. They require an understanding of limit behaviors, ratio tests, and conditions for convergence.
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Given

3.

The given series is 47-48+49-410+411-.......

We have to find the series is convergence or divergence.

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