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Prove the ratio test (Theorem 7.2.3a). What does this tell you if limn-co |#n+1/xn| exists?

Prove the ratio test (Theorem 7.2.3a). What does this tell you if limn-co |#n+1/xn| exists?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Ratio Test for Series Convergence** In mathematical analysis, the ratio test is a useful tool to determine the convergence or divergence of infinite series. The test involves analyzing the limit of the absolute value of the ratio of successive terms in the series. **Statement of the Ratio Test:** If \[ \left|\frac{x_{n+1}}{x_n}\right| < r \] for all sufficiently large \( n \) and some \( r < 1 \), then the series \( \sum x_n \) converges absolutely. Conversely, if \[ \left|\frac{x_{n+1}}{x_n}\right| \geq 1 \] for all sufficiently large \( n \), then the series \( \sum x_n \) diverges. The ratio test is particularly effective for series where the terms involve factorials, exponentials, or other rapidly changing sequences.
Transcribed Image Text:**Ratio Test for Series Convergence** In mathematical analysis, the ratio test is a useful tool to determine the convergence or divergence of infinite series. The test involves analyzing the limit of the absolute value of the ratio of successive terms in the series. **Statement of the Ratio Test:** If \[ \left|\frac{x_{n+1}}{x_n}\right| < r \] for all sufficiently large \( n \) and some \( r < 1 \), then the series \( \sum x_n \) converges absolutely. Conversely, if \[ \left|\frac{x_{n+1}}{x_n}\right| \geq 1 \] for all sufficiently large \( n \), then the series \( \sum x_n \) diverges. The ratio test is particularly effective for series where the terms involve factorials, exponentials, or other rapidly changing sequences.
**Prove the ratio test (Theorem 7.2.3a). What does this tell you if \( \lim_{n \to \infty} |x_{n+1}/x_{n}| \) exists?** The image presents a mathematical problem related to the ratio test, which is a method used in calculus to determine the convergence or divergence of an infinite series. - **Ratio Test Background:** - The ratio test involves taking the limit of the absolute value of the ratio of consecutive terms in a series. - If the limit is less than 1, the series converges absolutely. - If the limit is greater than 1, or infinite, the series diverges. - If the limit equals 1, the test is inconclusive. - **Objective:** - Provide a proof for the ratio test (Theorem 7.2.3a). - Interpret the implications of the existence of the limit \( \lim_{n \to \infty} |x_{n+1}/x_{n}| \). This task requires applying mathematical reasoning and knowledge of series to prove and understand the behavior of sequences based on the given conditions.
Transcribed Image Text:**Prove the ratio test (Theorem 7.2.3a). What does this tell you if \( \lim_{n \to \infty} |x_{n+1}/x_{n}| \) exists?** The image presents a mathematical problem related to the ratio test, which is a method used in calculus to determine the convergence or divergence of an infinite series. - **Ratio Test Background:** - The ratio test involves taking the limit of the absolute value of the ratio of consecutive terms in a series. - If the limit is less than 1, the series converges absolutely. - If the limit is greater than 1, or infinite, the series diverges. - If the limit equals 1, the test is inconclusive. - **Objective:** - Provide a proof for the ratio test (Theorem 7.2.3a). - Interpret the implications of the existence of the limit \( \lim_{n \to \infty} |x_{n+1}/x_{n}| \). This task requires applying mathematical reasoning and knowledge of series to prove and understand the behavior of sequences based on the given conditions.
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