Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problema 10.8.55

Capitolo 10, Problema 10.8.55

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)cos(1 / k⁹)

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First, write down the general term of the series: \(a_k = \cos\left(\frac{1}{k^9}\right)\).

Recall that for a series \(\sum a_k\) to converge, the terms \(a_k\) must approach zero as \(k\) approaches infinity. So, evaluate \(\lim_{k \to \infty} a_k = \lim_{k \to \infty} \cos\left(\frac{1}{k^9}\right)\).

Since \(\frac{1}{k^9} \to 0\) as \(k \to \infty\), use the continuity of cosine to find \(\lim_{k \to \infty} \cos\left(\frac{1}{k^9}\right) = \cos(0) = 1\).

Because the terms \(a_k\) do not approach zero (they approach 1), the necessary condition for convergence of the series is not met. Therefore, the series \(\sum_{k=1}^\infty \cos\left(\frac{1}{k^9}\right)\) diverges.

In conclusion, no further convergence tests are needed since the terms do not tend to zero, and the series diverges by the Test for Divergence (also called the nth-term test).

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Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity or oscillates.

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Percorso guidato

06:52

Convergence of an Infinite Series

Limit Comparison and Behavior of Terms

For a series to converge, its terms must approach zero as k approaches infinity. Analyzing the limit of the general term, such as cos(1/k⁹), helps determine if the terms tend to zero or a nonzero value, which is a necessary condition for convergence.

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07:45

Limit Comparison Test

Convergence Tests for Series

Various tests like the Comparison Test, Limit Comparison Test, and the Divergence Test help determine series convergence. Applying these tests to the given series involves comparing it to known convergent or divergent series or examining the limit of terms to justify convergence or divergence.

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07:51

Choosing a Convergence Test

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48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.

∑ (k = 2 to ∞) 1 / eᵏ

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13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{(√(4n⁴ + 3n))⁄(8n² + 1)}

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11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 0 to ∞) (3ᵏ⁺⁴) / (5ᵏ⁻²)

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