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Chapter 9: Problem 5

What tests are advisable if the series involves a factorial term?

Short Answer

Expert verified
Answer: The appropriate convergence tests to use for series involving factorial terms include the Ratio Test, the Root Test, the Direct Comparison Test, and the Limit Comparison Test. The Ratio and Root Tests are useful for various types of factorial series. The Direct Comparison Test can be helpful when there is a known convergent or divergent series to compare with. Lastly, the Limit Comparison Test compares two series and is suitable when the series have positive terms. Choosing the right test depends on the specific series under consideration.

Step by step solution

01

1. Identify the series with a factorial term

First, we should make sure that the series in question includes a factorial term. Factorials are usually denoted by n! (e.g., 3! = 3 脳 2 脳 1 = 6). The series should involve the sum of terms containing factorials, like 鈭�(n! / n^n) or 鈭�(n! / 2^n).
02

2. The Ratio Test

The Ratio Test is a common approach to testing convergency for series with factorial terms. Given the series 鈭慳_n, compute the limit L = lim (n鈫掆垶) (|a_(n+1) / a_n|). If L < 1, the series converges. If L > 1, the series diverges. If L = 1, the test is inconclusive.
03

3. The Root Test

The Root Test is another useful method when working with series that include a factorial term. Given the series 鈭慳_n, compute the limit M = lim (n鈫掆垶) (|a_n|^1/n). If M < 1, the series converges. If M > 1, the series diverges. If M = 1, the test is inconclusive.
04

4. The Direct Comparison Test

The Direct Comparison Test can sometimes help analyze series with factorials. If 0 鈮� a_n 鈮� b_n for all n, and the series 鈭慴_n converges, then the series 鈭慳_n also converges. Conversely, if the series 鈭慴_n diverges, then the series 鈭慳_n diverges as well.
05

5. The Limit Comparison Test

The Limit Comparison Test proposes to compare with another series. Let鈥檚 consider series 鈭慳_n and 鈭慴_n, which have positive terms, and calculate the limit C = lim (n鈫掆垶) (a_n / b_n). If 0 < C < 鈭�, then either both series converge or both diverge. In conclusion, advisable tests for series involving factorial terms include the Ratio Test, the Root Test, the Direct Comparison Test, and the Limit Comparison Test. It's crucial to choose the appropriate test depending on the specific series under consideration.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ratio Test
The Ratio Test is particularly handy when you're dealing with series that involve factorials. These are often characterized by very rapid growth or decline of terms. The test works by examining the ratio of successive terms in the series.
For a series \( \sum a_n \), calculate:
  • The limit \( L = \lim_{{n\to\infty}} \left|\frac{a_{n+1}}{a_n}\right| \).
  • If \( L < 1 \), the series converges. This tells you that the successive terms are getting smaller and smaller.
  • If \( L > 1 \) or is infinite, the series diverges. Here, terms are not diminishing fast enough to add up to a finite sum.
  • If \( L = 1 \), the test is inconclusive, and you might need another test to decide.
Factorials often appear in sequences where the Ratio Test effectively simplifies calculations by cancelling many terms. It's also quite effective for series in the form of \( \frac{n!}{n^n} \) or similar.
Root Test
The Root Test offers another route to evaluate convergence for series with factorials. It measures the nth root of the absolute value of the terms. This test is particularly suitable for series with terms of the form \( a_n^n \), which sometimes appear with factorials.
For a series \( \sum a_n \), determine:
  • The limit \( M = \lim_{n\to\infty} \left|a_n\right|^{1/n} \).
  • If \( M < 1 \), the series converges. It implies terms shrink rapidly enough in size.
  • If \( M > 1 \), the series diverges, indicating that terms do not shrink sufficiently.
  • If \( M = 1 \), the test offers no conclusion.
This test can be particularly powerful when dealing with exponential terms and sometimes when factorial terms reduce to such forms.
Direct Comparison Test
The Direct Comparison Test is a classic method to compare a series with another, simpler series that you already understand well. For dealing with factorial terms, this test can serve when a series can be naturally aligned with another known one.
To use the Direct Comparison Test, consider two series \( \sum a_n \) and \( \sum b_n \):
  • If \( 0 \leq a_n \leq b_n \) for all terms, and \( \sum b_n \) converges, then \( \sum a_n \) also converges.
  • On the flip side, if \( \sum b_n \) diverges, then \( \sum a_n \) diverges as well.
By manipulating inequalities and identifying bounds, you can often simplify complex factorial ensemble terms into manageable comparisons for practical convergence analysis.
Limit Comparison Test
The Limit Comparison Test is particularly useful for comparing series that do not so easily fit into a direct inequality, which is helpful with factorial terms. It works by checking the limit of the ratio of the terms of two series.
Here's how it's applied:
  • For two series \( \sum a_n \) and \( \sum b_n \) with positive terms, compute the limit: \[ C = \lim_{{n\to\infty}} \frac{a_n}{b_n} \].
  • If \( 0 < C < \infty \), then both series either converge or diverge together.
This test allows more flexibility compared to the Direct Comparison Test and can often be effective when dealing with series whose terms are defined by complex factorial structures. It aids in establishing behavior by aligning with a series whose convergence properties are known.

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Most popular questions from this chapter

Series of squares Prove that if \(\sum a_{k}\) is a convergent series of positive terms, then the series \(\Sigma a_{k}^{2}\) also converges.

Pick two positive numbers \(a_{0}\) and \(b_{0}\) with \(a_{0}>b_{0}\) and write out the first few terms of the two sequences \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}:\) $$a_{n+1}=\frac{a_{n}+b_{n}}{2}, \quad b_{n+1}=\sqrt{a_{n} b_{n}}, \quad \text { for } n=0,1,2 \dots$$ (Recall that the arithmetic mean \(A=(p+q) / 2\) and the geometric mean \(G=\sqrt{p q}\) of two positive numbers \(p\) and \(q\) satisfy \(A \geq G\). a. Show that \(a_{n}>b_{n}\) for all \(n\). b. Show that \(\left\\{a_{n}\right\\}\) is a decreasing sequence and \(\left\\{b_{n}\right\\}\) is an increasing sequence. c. Conclude that \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) converge. d. Show that \(a_{n+1}-b_{n+1}<\left(a_{n}-b_{n}\right) / 2\) and conclude that \(\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n} .\) The common value of these limits is called the arithmetic-geometric mean of \(a_{0}\) and \(b_{0},\) denoted \(\mathrm{AGM}\left(a_{0}, b_{0}\right)\). e. Estimate AGM(12,20). Estimate Gauss' constant \(1 / \mathrm{AGM}(1, \sqrt{2})\).

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty} \cos (\pi k)$$

Evaluate the limit of the following sequences. $$a_{n}=\frac{7^{n}}{n^{7} 5^{n}}$$

Consider the sequence \(\left\\{x_{n}\right\\}\) defined for \(n=1,2,3, \ldots\) by $$x_{n}=\sum_{k=n+1}^{2 n} \frac{1}{k}=\frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{2 n}$$ a. Write out the terms \(x_{1}, x_{2}, x_{3}\) b. Show that \(\frac{1}{2} \leq x_{n}<1,\) for \(n=1,2,3, \ldots\) c. Show that \(x_{n}\) is the right Riemann sum for \(\int_{1}^{2} \frac{d x}{x}\) using \(n\) subintervals. d. Conclude that \(\lim _{n \rightarrow \infty} x_{n}=\ln 2\)
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