Problem 59
Question
Which of the series in Exercises \(57-64\) converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty} \frac{(n !)^{n}}{\left(n^{n}\right)^{2}} $$
Step-by-Step Solution
Verified Answer
The series diverges by the Ratio Test as \( L = \infty \).
1Step 1: Understanding the Series
The given series is \( \sum_{n=1}^{\infty} \frac{(n !)^{n}}{(n^n)^2} \). Each term of the series is of the form \( a_n = \frac{(n !)^{n}}{(n^n)^2} \). We need to determine if this series converges or diverges.
2Step 2: Simplify the Term
Simplifying \( a_n \), we get:\[a_n = \frac{(n !)^{n}}{n^{2n}} = \left(\frac{n!}{n^n}\right)^{n}\]
3Step 3: Apply the Ratio Test
To apply the Ratio Test, evaluate the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_{n}} \right| \).First, express \( a_{n+1} \):\[ a_{n+1} = \left(\frac{(n+1)!}{(n+1)^{n+1}}\right)^{n+1} \]Calculate \( \frac{a_{n+1}}{a_{n}} \):\[\frac{a_{n+1}}{a_{n}} = \left(\frac{(n+1)!}{(n+1)^{n+1}} \div \frac{n!}{n^n}\right)^{n+1}\]This becomes:\[ \left(\frac{(n+1)^{n+1}}{n^{n} (n+1)}\right)^{n+1} = \left(\frac{(n+1)^n}{n^n}\right)^{n+1} = \left(1 + \frac{1}{n}\right)^{n(n+1)}\]
4Step 4: Calculate the Limit
Find \( L = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n(n+1)} \). As \( n \to \infty \), the expression becomes:\[\left((1 + \frac{1}{n})^n\right)^{n+1} \approx e^{n+1}\]Since \( e \) is a constant greater than 1, this limit approaches infinity: \( L = \infty \).
5Step 5: Conclusion with the Ratio Test
According to the Ratio Test, if \( L > 1 \), the series diverges. Because \( L \to \infty \), the series \( \sum_{n=1}^{\infty} \frac{(n !)^{n}}{(n^n)^2} \) diverges.
Key Concepts
Ratio testFactorialsInfinite seriesConvergence tests
Ratio test
The Ratio Test is a powerful tool used in evaluating the convergence or divergence of infinite series. When confronted with a complex series, this test simplifies the analysis. The main idea is to examine the limit of the ratio of successive terms in the series.
To apply the Ratio Test, we find the limit:\[L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]- If \( L < 1 \), the series converges absolutely.- If \( L > 1 \), the series diverges.- If \( L = 1 \), the test is inconclusive.
For example, when applied to our series \( \sum_{n=1}^{\infty} \frac{(n !)^{n}}{(n^n)^2} \), the limit \( L \) was found to be \( \infty \), indicating divergence because \( L > 1 \).
This method is incredibly useful for factorial and exponential functions, making our life much easier when deciding the nature of a series.
To apply the Ratio Test, we find the limit:\[L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]- If \( L < 1 \), the series converges absolutely.- If \( L > 1 \), the series diverges.- If \( L = 1 \), the test is inconclusive.
For example, when applied to our series \( \sum_{n=1}^{\infty} \frac{(n !)^{n}}{(n^n)^2} \), the limit \( L \) was found to be \( \infty \), indicating divergence because \( L > 1 \).
This method is incredibly useful for factorial and exponential functions, making our life much easier when deciding the nature of a series.
Factorials
Factorials are mathematical expressions used to represent the product of all positive integers up to a certain number, denoted by \( n! \). For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
The factorial is commonly found in series and sequences, often complicating matters when determining convergence or divergence. They grow very quickly, which is why series involving factorials tend to diverge unless controlled by other factors in the sequence.
In the given series \( \sum_{n=1}^{\infty} \frac{(n !)^{n}}{(n^n)^2} \), each term involves a factorial to the power of \( n \), resulting in extremely rapid growth. This kind of growth heavily influences the behavior of the series, often necessitating more rigorous tests like the Ratio Test for accurate determination of convergence or divergence.
The factorial is commonly found in series and sequences, often complicating matters when determining convergence or divergence. They grow very quickly, which is why series involving factorials tend to diverge unless controlled by other factors in the sequence.
In the given series \( \sum_{n=1}^{\infty} \frac{(n !)^{n}}{(n^n)^2} \), each term involves a factorial to the power of \( n \), resulting in extremely rapid growth. This kind of growth heavily influences the behavior of the series, often necessitating more rigorous tests like the Ratio Test for accurate determination of convergence or divergence.
Infinite series
An infinite series is the sum of an infinite sequence of numbers. It's written in the form \( \sum_{n=1}^{\infty} a_n \), where each \( a_n \) is a term in the series.
Infinite series can be convergent or divergent. A series converges if the sum approaches a finite number as the number of terms increases; it diverges if the sum grows without bound.
In our particular series \( \sum_{n=1}^{\infty} \frac{(n !)^{n}}{(n^n)^2} \), infinite series properties come into play as we explore its divergence using the Ratio Test. Understanding the behavior and properties of infinite series is crucial in calculus and mathematical analysis, as it allows us to deal with problems involving growth, approximation, and mathematical modeling.
Infinite series can be convergent or divergent. A series converges if the sum approaches a finite number as the number of terms increases; it diverges if the sum grows without bound.
In our particular series \( \sum_{n=1}^{\infty} \frac{(n !)^{n}}{(n^n)^2} \), infinite series properties come into play as we explore its divergence using the Ratio Test. Understanding the behavior and properties of infinite series is crucial in calculus and mathematical analysis, as it allows us to deal with problems involving growth, approximation, and mathematical modeling.
Convergence tests
Convergence tests are a set of strategies and methods used to determine whether an infinite series converges or diverges. Some of the most common convergence tests include:
- Ratio Test: Useful for series involving factorials, powers, and exponential terms.
- Root Test: Similar to the ratio test but uses the \( n \)th root instead of the ratio.
- Integral Test: Compares a series to an improper integral.
- Comparison Test: Uses a comparison with another series whose convergence behavior is known.
- Alternating Series Test: Used specifically for series with alternating terms.
Other exercises in this chapter
Problem 59
Suppose that \(a_{n}>0\) and \(b_{n}>0\) for \(n \geq N(N\) an integer). If \(\lim _{n \rightarrow \infty}\left(a_{n} / b_{n}\right)=\infty\) and \(\sum a_{n}\)
View solution Problem 59
In Exercises \(57 - 82 ,\) use any method to determine whether the series converges or diverges. Give reasons for your answer. $$ \sum _ { n = 1 } ^ { \infty }
View solution Problem 59
Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\sqrt[n]{10 n} $$
View solution Problem 59
Which series in Exercises \(53-76\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$\sum_{n=0}^{\infty} e^{-2
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