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
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85f1beef-b59a-44ac-a29d-5ba6518650e4%2F982c7f4c-c447-420f-a457-52e8ad6fb44f%2F5itvtrl_processed.png&w=3840&q=75)
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.
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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