Ch. 10 - Infinite Sequences and Series

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 10 - Infinite Sequences and Series

Problema 10.6.26

Capitolo 10, Problema 10.6.26

Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (ⁿ√10)]

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Identify the given series: \( \sum_{n=1}^{\infty} (-1)^{n+1} \sqrt[n]{10} \). This is an alternating series because of the factor \( (-1)^{n+1} \).

Check the absolute convergence by considering the series of absolute values: \( \sum_{n=1}^{\infty} \left| (-1)^{n+1} \sqrt[n]{10} \right| = \sum_{n=1}^{\infty} \sqrt[n]{10} \).

Analyze the behavior of the terms \( \sqrt[n]{10} = 10^{1/n} \). As \( n \to \infty \), \( 10^{1/n} \to 1 \), so the terms do not approach zero.

Since the terms of the absolute value series do not approach zero, the series \( \sum_{n=1}^{\infty} \sqrt[n]{10} \) diverges, so the original series does not converge absolutely.

Next, apply the Alternating Series Test to the original series: check if the terms \( \sqrt[n]{10} \) decrease monotonically to zero. Since they approach 1, not zero, the Alternating Series Test fails, so the series does not converge conditionally either.

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Absolute Convergence

A series ∑a_n converges absolutely if the series of absolute values ∑|a_n| converges. Absolute convergence guarantees convergence regardless of the order of terms and is a stronger form of convergence than conditional convergence.

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Choosing a Convergence Test

Conditional Convergence

A series converges conditionally if it converges, but does not converge absolutely. This means ∑a_n converges, but ∑|a_n| diverges. Alternating series often exhibit conditional convergence, relying on the alternating signs and decreasing terms.

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Choosing a Convergence Test

Root Test for Convergence

The root test uses the limit L = lim (n→∞) ⁿ√|a_n| to determine convergence: if L < 1, the series converges absolutely; if L > 1, it diverges; if L = 1, the test is inconclusive. This test is especially useful for series involving nth roots.

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Root Test

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Series with Geometric Terms

In Exercises 7–14, write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges.

∑ (from n = 0 to ∞) [(1 / 2ⁿ) + ((-1)ⁿ / 5ⁿ)]

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Finding Terms of a Sequence

Each of Exercises 1–6 gives a formula for the nth term aₙ of a sequence {aₙ}. Find the values of a₁, a₂, a₃, and a₄.

aₙ = 2 + (-1)ⁿ

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Determining Convergence or Divergence

Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

∑ (from n=1 to ∞) (1 + 1/n)ⁿ

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Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.

∑ (from n = 0 to ∞) e^(−2n)

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Does the series

∑ (from n=1 to ∞) (1/n − 1/n²)

converge or diverge? Justify your answer.

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