Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Question
**Topic: Convergence or Divergence of a Series** **Objective:** Determine if the following series converges or diverges: \[ \sum_{n=1}^{\infty} \frac{7n^2 + 6n - 3}{7n^2 + 4} \] **Explanation:** To determine if the series \[ \sum_{n=1}^{\infty} \frac{7n^2 + 6n - 3}{7n^2 + 4} \] converges or diverges, we can analyze the behavior of the general term as \( n \) tends to infinity. This can be done using various techniques such as the Limit Comparison Test, the Ratio Test, or by evaluating the limit analytically. Firstly, let's simplify the given term: \[ \frac{7n^2 + 6n - 3}{7n^2 + 4} \] For large values of \( n \), the highest degree terms \( 7n^2 \) in both the numerator and the denominator will dominate. So, we can approximate the given term by considering only these leading terms: \[ \frac{7n^2}{7n^2} = 1 \] Thus, as \( n \) tends to infinity, \(\frac{7n^2 + 6n - 3}{7n^2 + 4} \approx 1\). Since the general term does not approach zero but rather approaches 1, the series does not satisfy the necessary condition for convergence of an infinite series. Specifically, a series \(\sum a_n\) converges only if the sequence \(a_n\) approaches 0 as \(n\) approaches infinity. Therefore, we conclude that the series diverges.
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Transcribed Image Text:**Topic: Convergence or Divergence of a Series** **Objective:** Determine if the following series converges or diverges: \[ \sum_{n=1}^{\infty} \frac{7n^2 + 6n - 3}{7n^2 + 4} \] **Explanation:** To determine if the series \[ \sum_{n=1}^{\infty} \frac{7n^2 + 6n - 3}{7n^2 + 4} \] converges or diverges, we can analyze the behavior of the general term as \( n \) tends to infinity. This can be done using various techniques such as the Limit Comparison Test, the Ratio Test, or by evaluating the limit analytically. Firstly, let's simplify the given term: \[ \frac{7n^2 + 6n - 3}{7n^2 + 4} \] For large values of \( n \), the highest degree terms \( 7n^2 \) in both the numerator and the denominator will dominate. So, we can approximate the given term by considering only these leading terms: \[ \frac{7n^2}{7n^2} = 1 \] Thus, as \( n \) tends to infinity, \(\frac{7n^2 + 6n - 3}{7n^2 + 4} \approx 1\). Since the general term does not approach zero but rather approaches 1, the series does not satisfy the necessary condition for convergence of an infinite series. Specifically, a series \(\sum a_n\) converges only if the sequence \(a_n\) approaches 0 as \(n\) approaches infinity. Therefore, we conclude that the series diverges.
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