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 \(a_{n}=\frac{1}{1+x} \frac{2}{2+x} \cdots \frac{n}{n+x} \frac{1}{n}=\frac{(n-1) !}{(1+x)(2+x) \cdots(n+x)} .\) Show that \(a_{2 n} / a_{n} \leq e^{-x / 2} / 2 .\) For which \(x>0\) does the generalized ratio test imply convergence of \(\sum_{n=1}^{\infty} a_{n} ?\) (Hint: Write \(2 a_{2 n} / a_{n}\) as a product of \(n\) factors each smaller than \(1 /(1+x /(2 n))\)

Short answer

The series converges for \( x > 2 \ln(2) \).

Step-by-step solution

Define the Expression for a_n

The given sequence is defined as follows: You start with \( a_n = \frac{1}{1+x} \frac{2}{2+x} \cdots \frac{n}{n+x} \frac{1}{n} = \frac{(n-1)!}{(1+x)(2+x) \cdots (n+x)} \). Notice that \( a_n \) is essentially a ratio of factorials.

Express a_{2n} in Terms of a_n

Next, find the expression for \( a_{2n} \). Using the definition, \( a_{2n} = \frac{(2n-1)!}{(1+x)(2+x) \cdots (2n+x)} \).

Compute the Ratio a_{2n}/a_n

To establish the ratio, divide the expression for \( a_{2n} \) by \( a_n \):\[\frac{a_{2n}}{a_n} = \frac{(2n-1)!}{(n-1)!} \cdot \frac{(1+x)(2+x)\cdots(n+x)}{(n+1+x)(n+2+x)\cdots(2n+x)}\]

Approximate each Factor in the Product

For each \( k \) from \( n+1 \) to \( 2n \), approximate \( \frac{k}{k+x} \approx \frac{1}{1+x/(2n)} \) using the hint provided, as these are the factors that will contribute to the product simplifying.

Estimate the Product

Combine the approximations to obtain:\[\frac{a_{2n}}{a_n} \approx \prod_{k=1}^{n} \frac{1}{1+x/(2n)}\]This is approximately \( \left( \frac{1}{1+x/(2n)} \right)^n \). For large \( n \), \( \left(1 + \frac{x}{2n}\right)^n \approx e^{-x/2} \).

Final Expression for the Limit

Therefore, as \( n \to \infty \), \( \frac{a_{2n}}{a_n} \leq e^{-x/2} \). Thus, we write:\[\lim_{n \to \infty} \frac{a_{2n}}{a_n} = e^{-x/2}.\]

Determine Values of x for Convergence

For convergence, according to the test, we need \( e^{-x/2} < \frac{1}{2} \). This implies that \( x > 2 \ln(2) \approx 1.386.\)

Key concepts

Convergence of Series

Understanding the convergence of series is crucial in determining the behavior of infinite sums. When a series converges, it means the terms add up to a specific value as you include more and more terms. In mathematical terms, a series \( \sum a_n \) is said to converge if the sequence of partial sums \( S_n = a_1 + a_2 + \ldots + a_n \) approaches a limit as \( n \to \infty \).
For a particular class of series, the generalized ratio test provides a tool for convergence analysis when other tests like the standard ratio or root test are inadequate. It explores the behavior of terms at different growth rates to ascertain convergence or divergence. In our exercise, convergence depends on the limit behavior of the "double" sequence terms, analyzed using this specialized form of the ratio test.
Key points to remember:

• A series converges if its terms form a sequence of partial sums approaching a finite limit.
• The generalized ratio test involves comparing the initial term sequence with a transformed sequence double its index.
• Understanding when conventional tests become less effective paves the way for using more powerful alternatives like the generalized ratio test.

Factorials in Sequences

Factorials often appear in sequences and series because they provide a structured and progressive way to express terms through rapid growth. Here, the factorial \( (n-1)! \) expresses the product of an integer sequence, which is significant in calculating each \( a_n \).
Factorials serve to represent how quickly sequences grow, which is instrumental in determining the behavior of series when the terms involve increasingly large numbers, as they typically do in ratios and products.
In our exercise:

• The sequence \( a_n \) is defined using factorials in the numerator, showcasing a polynomial growth progression.
• The challenge lies in comparing these factorials expanded into sequences, like \( (2n-1)! \) versus sequences embedded in the denominator, like \( (1+x)\cdots(n+x) \).
• The generalized ratio test exploits factorials' predictable structure to establish convergence or divergence.

Approximation Methods

When dealing with complicated series, approximation methods can simplify calculations, especially as terms grow large. Approximations simplify complex mathematical expressions into more manageable terms that maintain the essence of the original function's behavior.
In our exercise, one key approximation involved linking the fraction \( \frac{k}{k+x} \) to \( \frac{1}{1+x/(2n)} \). This helps simplify the products for ease of understanding and clearer insights into convergence properties.
Remember:

• Approximations transform unwieldy fractions into terms easier to handle, especially in limit calculations.
• In large \( n \), terms like \( (1 + \frac{x}{2n})^n \) closely resemble expressions involving the exponential function \( e^{-x/2} \).
• These techniques can guide reasoning about a series' behavior in limit calculations and convergence tests.

Limit Calculations

Limit calculations help in evaluating the asymptotic behavior of sequences as they grow indefinitely. Approaching the limit \( n \to \infty \) means observing how expressions change when their terms and factors increase without bound, which is fundamental in series convergence tests.
In our exercise, achieving convergent insights through the limits like \( \lim _{n \to \infty} \frac{a_{2n}}{a_n} \) demonstrates the significance of bounds in series analysis. It's essential to understand the conditions ensuring convergence based on the results of the limit functions.
Guidance includes:

• Ensure comprehension of limits involving exponential forms, as these frequently underpin convergence conclusions.
• Apply limit laws to determine the final behavior of sequence ratios, such as transitioning from \( \prod_{k=1}^{n} \) terms to exponential approximations.
• Use precise limit calculations to satisfy generalized test conditions, revealing hidden insights about the series' behavior.