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Which of the following is an example of a divergent infinite series whose general term tends to zero? 1 Α.Σ n=1 Β.Σ n=1 c.Σ.20 n=1 9 Ο.Σ η 10 n=1 - 9 η 10 n 9 1+ n 10 ole 9 10

Which of the following is an example of a divergent infinite series whose general term tends to zero? 1 Α.Σ n=1 Β.Σ n=1 c.Σ.20 n=1 9 Ο.Σ η 10 n=1 - 9 η 10 n 9 1+ n 10 ole 9 10

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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**Title: Infinite Series and Convergence** **Question:** Which of the following is an example of a divergent infinite series whose general term tends to zero? **Options:** A. \( \sum_{n=1}^{\infty} \frac{1}{n^{\frac{9}{10}}} \) B. \( \sum_{n=1}^{\infty} \frac{n^{\frac{9}{10}}}{1+n^{\frac{9}{10}}} \) C. \( \sum_{n=1}^{\infty} 2^{-n} \) D. \( \sum_{n=1}^{\infty} n^{\frac{9}{10}} \) **Explanation:** When dealing with infinite series, it's crucial to determine whether the series converges or diverges. Convergence means that the sum of the infinite series approaches a finite number, while divergence means the sum grows without bound (or oscillates without settling to a particular value). To understand why a series might converge or diverge, we consider the general term of the series and its behavior as \( n \) approaches infinity. For a series \( \sum_{n=1}^{\infty} a_n \), convergence is typically evaluated using tests like the Ratio Test, Root Test, or comparison with known convergent/divergent series. ### Analysis of Each Option: - **Option A:** \( \sum_{n=1}^{\infty} \frac{1}{n^{\frac{9}{10}}} \) - Here, the general term \( \frac{1}{n^{\frac{9}{10}}} \) tends to zero as \( n \) approaches infinity. - Despite the general term tending to zero, examining the p-series test (where \( p = \frac{9}{10} < 1 \)) tells us this series diverges since \( p \) is less than 1. - **Option B:** \( \sum_{n=1}^{\infty} \frac{n^{\frac{9}{10}}}{1+n^{\frac{9}{10}}} \) - Here, the general term does not tend to zero because even as \( n \) increases, \( \frac{n^{\frac{9}{10}}}{1+n^{\frac{9}{10}}} \)
Transcribed Image Text:**Title: Infinite Series and Convergence** **Question:** Which of the following is an example of a divergent infinite series whose general term tends to zero? **Options:** A. \( \sum_{n=1}^{\infty} \frac{1}{n^{\frac{9}{10}}} \) B. \( \sum_{n=1}^{\infty} \frac{n^{\frac{9}{10}}}{1+n^{\frac{9}{10}}} \) C. \( \sum_{n=1}^{\infty} 2^{-n} \) D. \( \sum_{n=1}^{\infty} n^{\frac{9}{10}} \) **Explanation:** When dealing with infinite series, it's crucial to determine whether the series converges or diverges. Convergence means that the sum of the infinite series approaches a finite number, while divergence means the sum grows without bound (or oscillates without settling to a particular value). To understand why a series might converge or diverge, we consider the general term of the series and its behavior as \( n \) approaches infinity. For a series \( \sum_{n=1}^{\infty} a_n \), convergence is typically evaluated using tests like the Ratio Test, Root Test, or comparison with known convergent/divergent series. ### Analysis of Each Option: - **Option A:** \( \sum_{n=1}^{\infty} \frac{1}{n^{\frac{9}{10}}} \) - Here, the general term \( \frac{1}{n^{\frac{9}{10}}} \) tends to zero as \( n \) approaches infinity. - Despite the general term tending to zero, examining the p-series test (where \( p = \frac{9}{10} < 1 \)) tells us this series diverges since \( p \) is less than 1. - **Option B:** \( \sum_{n=1}^{\infty} \frac{n^{\frac{9}{10}}}{1+n^{\frac{9}{10}}} \) - Here, the general term does not tend to zero because even as \( n \) increases, \( \frac{n^{\frac{9}{10}}}{1+n^{\frac{9}{10}}} \)
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