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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.8.49
Capitolo 10, Problema 10.8.49

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)(⁵√k) / ⁵√(k⁷ + 1)

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First, rewrite the general term of the series to better understand its behavior: the term is given by \(\frac{\sqrt[5]{k}}{\sqrt[5]{k^{7} + 1}}\).
Simplify the expression inside the fifth root in the denominator by factoring out \(k^{7}\): \(\sqrt[5]{k^{7} + 1} = \sqrt[5]{k^{7}(1 + \frac{1}{k^{7}})}\).
Use the property of roots to separate the terms: \(\sqrt[5]{k^{7}(1 + \frac{1}{k^{7}})} = \sqrt[5]{k^{7}} \cdot \sqrt[5]{1 + \frac{1}{k^{7}}} = k^{\frac{7}{5}} \cdot \sqrt[5]{1 + \frac{1}{k^{7}}}\).
Rewrite the original term as \(\frac{k^{\frac{1}{5}}}{k^{\frac{7}{5}} \cdot \sqrt[5]{1 + \frac{1}{k^{7}}}} = \frac{1}{k^{\frac{6}{5}} \cdot \sqrt[5]{1 + \frac{1}{k^{7}}}}\).
As \(k\) approaches infinity, \(\sqrt[5]{1 + \frac{1}{k^{7}}}\) approaches 1, so the term behaves like \(\frac{1}{k^{\frac{6}{5}}}\). Use the p-series test to determine convergence: since \(\frac{6}{5} > 1\), the series converges.

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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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Comparison Test for Series

The Comparison Test involves comparing the given series to a known benchmark series with positive terms. If the given series is smaller than a convergent series or larger than a divergent series, conclusions about its convergence or divergence can be drawn.
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Direct Comparison Test

Asymptotic Behavior and Simplification of Terms

Analyzing the behavior of terms for large indices (k → ∞) helps simplify complex expressions. Approximating dominant terms allows us to compare the series to simpler p-series or geometric series, facilitating the application of convergence tests.
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Domanda del libro di testo

55–70. More sequences

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


aₙ = (−1)ⁿ ⁿ√n

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Domanda del libro di testo

13–52. Limits of sequences

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


{(3ⁿ⁺¹ + 3)⁄3ⁿ}

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Domanda del libro di testo

55–70. More sequences

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


{tan⁻¹n / n}

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Domanda del libro di testo

13–52. Limits of sequences

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


{(1 + (2 / n))ⁿ}

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Domanda del libro di testo

1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.

∑ (from k = 1 to ∞) (k² / (k⁴ + k³ + 1))

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54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.


63. ∑ (k = 1 to ∞) 1 / ((k + p)(k + p + 1)), where p is a positive integer

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