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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 10.5.50
Capitolo 10, Problema 10.5.50

40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 2 to ∞) (5lnk) / k

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Identify the series given: \( \sum_{k=2}^{\infty} \frac{5 \ln k}{k} \). We want to determine if this series converges or diverges.
Since the terms involve \( \ln k \) and \( k \), consider using the Integral Test, which is suitable for series with positive, continuous, and decreasing terms for large \( k \).
Set up the corresponding integral for the Integral Test: \( \int_{2}^{\infty} \frac{5 \ln x}{x} \, dx \).
Evaluate the integral by using substitution: let \( u = \ln x \), then \( du = \frac{1}{x} dx \), so the integral becomes \( 5 \int u \, du \).
Determine whether the integral converges or diverges by evaluating the limit as the upper bound approaches infinity. If the integral converges, the series converges; if it diverges, the series diverges.

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Convergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely. Determining convergence involves analyzing the behavior of the terms and applying appropriate tests to see if the partial sums stabilize.
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Convergence of an Infinite Series

Comparison and Limit Comparison Tests

These tests compare the given series to a known benchmark series. The Comparison Test checks if terms are smaller or larger than a convergent or divergent series, while the Limit Comparison Test uses the limit of the ratio of terms to determine convergence behavior.
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Integral Test

The Integral Test relates the convergence of a series to the convergence of an improper integral of a related function. If the integral of f(x) from some point to infinity converges, then the series ∑ f(k) also converges, provided f is positive, continuous, and decreasing.
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Integral Test
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45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 1 to ∞) (−1)ᵏ · k / (2k + 1)

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Growth rates of sequences

Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.


{n¹⁰⁰⁰ / 2ⁿ}

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Is it possible for an alternating series to converge absolutely but not conditionally?

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Periodic doses

Suppose you take 200 mg of an antibiotic every 6 hr. The half-life of the drug (the time it takes for half of the drug to be eliminated from your blood) is 6 hr. Use infinite series to find the long-term (steady-state) amount of antibiotic in your blood. You may assume the steady-state amount is finite.

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9–15. Geometric sums Evaluate each geometric sum.


{Use of Tech}∑ k = 0 to 9(−3/4)ᵏ

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61–66. Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges.


4 + 0.9 + 0.09 + 0.009 + ⋯  

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