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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 10.5.15
Chapter 10, Problem 10.5.15

Using the Root Test
In Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.
∑(from n=1 to ∞) [(-1)ⁿ (1 − 1/n)ⁿ^²]
(Hint: lim (n→∞) (1 + x/n)ⁿ = eˣ)

Verified step by step guidance
1
Identify the general term of the series: \(a_n = (-1)^n \left(1 - \frac{1}{n}\right)^{n^2}\).
Apply the Root Test, which involves computing the limit \(L = \lim_{n \to \infty} \sqrt[n]{|a_n|}\). Since \(|a_n| = \left(1 - \frac{1}{n}\right)^{n^2}\), we have \(\sqrt[n]{|a_n|} = \left(\left(1 - \frac{1}{n}\right)^{n^2}\right)^{\frac{1}{n}} = \left(1 - \frac{1}{n}\right)^n\).
Evaluate the limit \(L = \lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^n\). Using the hint, recall that \(\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x\). Here, \(x = -1\), so \(L = e^{-1}\).
Interpret the Root Test result: If \(L < 1\), the series converges absolutely; if \(L > 1\), it diverges; if \(L = 1\), the test is inconclusive. Since \(e^{-1} < 1\), the series converges absolutely.
Conclude that the series \(\sum_{n=1}^\infty (-1)^n \left(1 - \frac{1}{n}\right)^{n^2}\) converges absolutely by the Root Test.

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Key Concepts

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

Root Test for Series Convergence

The Root Test determines the convergence of a series by examining the nth root of the absolute value of its terms. Specifically, if the limit of the nth root is less than 1, the series converges absolutely; if greater than 1, it diverges; and if equal to 1, the test is inconclusive.
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Root Test

Absolute Convergence

A series converges absolutely if the series of the absolute values of its terms converges. Absolute convergence implies convergence regardless of the sign of terms, which is crucial when applying tests like the Root Test that consider absolute values.
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Choosing a Convergence Test

Limit Definition of the Exponential Function

The limit lim (n→∞) (1 + x/n)^n = e^x defines the exponential function and is used to evaluate limits involving expressions raised to the nth power. This concept helps simplify the limit in the Root Test when terms involve expressions like (1 - 1/n) raised to powers involving n.
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Exponential Functions
Related Practice
Textbook Question

Are there any values of x for which ∑ (from n=1 to ∞) (1 / nˣ) converges? Give reasons for your answer.

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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 = 2 to ∞) [(-1)ⁿ⁺¹ (1 / ln n)]

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Textbook Question

Using the Ratio Test

In Exercises 1–8, use the Ratio Test to determine whether each series converges absolutely or diverges.

∑(from n=1 to ∞) [(-1)ⁿ (n + 2) / 3ⁿ]

Textbook Question

Recursively Defined Sequences

In Exercises 101–108, assume that each sequence converges and find its limit.

a₁ = 2,aₙ₊₁ = 72 / (1 + aₙ)

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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 = nπ cos(nπ)

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Textbook Question

Uniqueness of limits Prove that limits of sequences are unique. That is, show that if L₁ and L₂ are numbers such that aₙ → L₁ and aₙ → L₂, then L₁ = L₂.

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