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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblema 10.5.15
Capitolo 10, Problema 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)

Guida verificata passo dopo passo
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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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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Percorso guidato
07:15
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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Percorso guidato
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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
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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 ∞) [4ⁿ / (3n)ⁿ]

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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]

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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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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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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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Convergence and Divergence

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

aₙ = (n + 3) / (n² + 5n + 6)

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