Question

The following advanced exercises use a generalized ratio test to determine convergence of some series that arise in particular applications when tests in this chapter, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if \(\lim _{n \rightarrow \infty} \frac{a_{2 n}}{a_{n}}<1 / 2\), then \(\sum a_{n}\) converges, while if \(\lim _{n \rightarrow \infty} \frac{a_{2 n+1}}{a_{n}}>1 / 2\), then \(\sum a_{n}\) diverges. Let \(a_{n}=\frac{1}{4} \frac{3}{6} \frac{5}{8} \cdots \frac{2 n-1}{2 n+2}=\frac{1 \cdot 3-5 \cdots(2 n-1)}{2^{n}(n+1) !} .\) Explain why the ratio test cannot determine convergence of \(\sum_{n=1}^{\infty} a_{n} .\) Use the fact that \(1-1 /(4 k)\) is increasing \(k\) to estimate \(\lim _{n \rightarrow \infty} \frac{a_{2 n}}{a_{n}}\).

Short answer

The ratio test is inconclusive, but using the generalized ratio test, \( \sum a_n \) converges.

Step-by-step solution

Setup the Ratio Test

To apply the Ratio Test, we consider the limit: \[ \lim_{n \to \infty} \frac{|a_{n+1}|}{|a_{n}|} \]. For the series \( a_n = \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2^n (n+1)!} \), the ratio test requires us to find this limit and check if it is less than 1, equal to 1, or greater than 1 to determine convergence.

Apply the Ratio Test

Substitute \( a_{n+1} = \frac{1 \cdot 3 \cdot 5 \cdots (2n+1)}{2^{n+1} (n+2)!} \) into the ratio test formula and simplify:\[ \frac{1 \cdot 3 \cdot 5 \cdots (2n+1)}{2^{n+1} (n+2)!} \div \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2^n (n+1)!} \].This becomes:\[ \frac{(2n+1) \cdot (n+1)! \cdot 2^n}{2^{n+1} (n+2)! \cdot (2n-1)\cdots 1} = \frac{2n+1}{2(n+2)} \].Thus, the limit becomes:\[ \lim_{n \to \infty} \frac{2n+1}{2(n+2)} = \frac{1}{1} \].

Conclusion from Ratio Test

Since the ratio test limit \( \frac{1}{1} = 1 \), the ratio test is inconclusive in determining the convergence of \( \sum a_n \).

Analyze the Generalized Ratio Test

We consider the generalized ratio test for \( \sum a_n \). Calculate:\[ \lim_{n \to \infty} \frac{a_{2n}}{a_n} = \frac{\frac{1 \cdot 3 \cdot 5 \cdots (4n-1)}{2^{2n} (2n+1)!}}{\frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2^n (n+1)!}} \].This simplifies to:\[ \frac{(2n)(2n-1) \cdots (n+1)}{2^n (2n+1) \cdots (4n-1)} \].

Estimation and Conclusion

Estimate using \(1 - \frac{1}{4k}\) as given, which helps us understand that each fraction's numerator is always a bit less than its denominator. Thus,\[ \lim_{n \to \infty} \frac{a_{2n}}{a_n} < \frac{1}{2} \].By the generalized ratio test, the series \( \sum a_n \) converges as the condition \( \lim_{n \to \infty} \frac{a_{2n}}{a_n} < \frac{1}{2} \) is satisfied.

Key concepts

Generalized Ratio Test

The Generalized Ratio Test is an extension of the traditional Ratio Test used to assess the convergence of infinite series. It provides additional criteria for determining whether a series converges or diverges, especially when the standard ratio test ends up being inconclusive.
This test involves analyzing the limit of the ratio of every second term in a sequence. For a series \( \sum a_n \), it checks the limit \( \lim_{n \to \infty} \frac{a_{2n}}{a_n} \). If this limit is less than \( \frac{1}{2} \), the series converges. Conversely, if it is greater than \( \frac{1}{2} \), the series diverges.
This test is particularly powerful when dealing with series involving complex factorials or alternating products, where simple convergence tests may not apply. In problems where the sequence changes its form irregularly, this additional analysis helps in providing crucial insights.

Series Convergence

Series convergence refers to whether the sum of an infinite series approaches a definite value. It is a fundamental concept in calculus and mathematical analysis, crucial for evaluating functions, determining the size of errors in approximations, and understanding various natural phenomena.
There are several tests used to determine series convergence, including the Ratio Test, the Root Test, and the Comparison Test, among others. Each test has its conditions and suitability, which typically depend on the structure of the series being analyzed.
The Ratio Test is one common method, where we look at the ratio \( \frac{|a_{n+1}|}{|a_n|} \) as \( n \to \infty \) to determine convergence. If this ratio is less than 1, the series converges; if greater than 1, it diverges; and if exactly 1, the test is inconclusive, requiring a different convergence strategy, like the generalized ratio test used in this context.

Infinite Series Analysis

Infinite Series Analysis is a crucial aspect of understanding patterns and behaviors in mathematics. It involves employing various tests and techniques to determine the nature of a series, whether it converges, diverges, or behaves in some periodic manner.
The study of infinite series is not only about finding the sum of an endless series of terms but also about understanding the growth rate and the limit behavior as the number of terms increases. It delves into concepts like conditional convergence, where partial sums of alternating series can yield convergence even when individual components diverge.
Techniques like the generalized ratio test, root test, and integration tests form a central toolkit for analysts. Their primary objective is to simplify often complex sequences into more manageable forms, thereby leading to interpretations that aid practical applications, from physics to finance.