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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.5.60
Chapter 10, Problem 10.5.60

Convergence or Divergence
Which of the series in Exercises 57–64 converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ (n!)ⁿ] / [n^(n²)]

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1
Identify the given series: \( \sum_{n=1}^{\infty} \frac{(-1)^n (n!)^n}{n^{n^2}} \). We want to determine whether this series converges or diverges.
Since the series involves factorials and powers raised to powers, consider using the Root Test, which is effective for series with terms raised to the \( n \)-th power. The Root Test uses the limit \( L = \lim_{n \to \infty} \sqrt[n]{|a_n|} \), where \( a_n = \frac{(-1)^n (n!)^n}{n^{n^2}} \).
Calculate \( \sqrt[n]{|a_n|} = \sqrt[n]{\frac{(n!)^n}{n^{n^2}}} = \frac{n!}{n^{n}} \) because \( \sqrt[n]{(n!)^n} = n! \) and \( \sqrt[n]{n^{n^2}} = n^{n} \).
Analyze the behavior of \( \frac{n!}{n^n} \) as \( n \to \infty \). Use Stirling's approximation for factorials: \( n! \approx \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n \). Substitute this approximation to understand the limit.
Evaluate the limit \( L = \lim_{n \to \infty} \frac{n!}{n^n} \approx \lim_{n \to \infty} \frac{\sqrt{2 \pi n} \left( \frac{n}{e} \right)^n}{n^n} = \lim_{n \to \infty} \sqrt{2 \pi n} \left( \frac{1}{e} \right)^n \). Since \( \left( \frac{1}{e} \right)^n \) tends to zero exponentially, \( L = 0 \). According to the Root Test, if \( L < 1 \), the series converges absolutely.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Convergence and Divergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely; otherwise, it diverges. Determining convergence involves analyzing the behavior of the terms and applying appropriate tests to assess whether the series sums to a finite value.
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Convergence of an Infinite Series

Ratio Test

The Ratio Test evaluates the limit of the absolute value of the ratio of consecutive terms in a series. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive. This test is especially useful for series involving factorials and exponential terms.
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Factorials and Exponential Growth

Factorials (n!) grow faster than exponential functions as n increases, which significantly affects the behavior of series terms. Understanding how factorials compare to powers and exponentials helps in applying convergence tests and predicting whether terms shrink quickly enough for the series to converge.
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Related Practice
Textbook Question

Error Estimates

The approximation eˣ = 1 + x + (x² / 2) is used when x is small. Use the Remainder Estimation Theorem to estimate the error when |x| < 0.1.

Textbook Question

Finding Taylor Polynomials

In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.

f(x) = √(1 − x),a = 0

Textbook Question

Determining Convergence or Divergence

In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (1 / n^(3/2))]

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Textbook Question

Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = 2 + (0.1)ⁿ

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Textbook Question

In Exercises 53–56, determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001.

∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (n / (n² + 1))]

Textbook Question

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) 1 / (2√n + ³√n)

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