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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
Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.8.29
Capitolo 10, Problema 10.8.29

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (10ᵏ + 1) / k¹⁰

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Identify the general term of the series: \(a_k = \frac{10^k + 1}{k^{10}}\).
Observe the behavior of the numerator and denominator separately: the numerator grows exponentially as \$10^k$, while the denominator grows polynomially as $k^{10}$.
Recall that exponential growth dominates polynomial growth, so \$10^k$ grows much faster than $k^{10}$ as \(k \to \infty\).
Apply the Divergence Test (also known as the Test for Divergence) by checking the limit of \(a_k\) as \(k\) approaches infinity: calculate \(\lim_{k \to \infty} \frac{10^k + 1}{k^{10}}\).
Since the numerator grows exponentially and the denominator polynomially, this limit does not approach zero; therefore, by the Divergence Test, the series diverges.

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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.
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Convergence of an Infinite Series

Comparison Test

The Comparison Test involves comparing a given series to a second series with known convergence behavior. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges.
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Direct Comparison Test

Behavior of Exponential vs. Polynomial Terms

Exponential terms like 10^k grow much faster than polynomial terms like k^10. When analyzing series terms involving both, the exponential growth dominates, often causing divergence despite the polynomial denominator.
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Graphs of Exponential Functions
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9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

∑ (from k = 1 to ∞) ((-1)ᵏ) / (k!)

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46–53. Decimal expansions

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).


49.0.037̅ = 0.037037…

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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.


57. ∑ (k = 1 to ∞) 1 / ((k + 6)(k + 7))

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32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞) (−1)ᵏ k (2ᵏ⁺¹ / (9ᵏ − 1))

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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 ∞) ((−1)ᵏ⁺¹) / (√2ᵏ + lnk)

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38–39. Examining a series two ways Determine whether the following series converge using either the Comparison Test or the Limit Comparison Test. Then use another method to check your answer.

39. ∑ (k = 1 to ∞) 1 / (k² + 2k + 1)

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