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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 10.8.81
Capitolo 10, Problema 10.8.81

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


∑ (from k = 1 to ∞)(1 / √(k + 2) – 1 / √k)

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Recognize that the series is given by \( \sum_{k=1}^{\infty} \left( \frac{1}{\sqrt{k+2}} - \frac{1}{\sqrt{k}} \right) \). This is a series whose terms are differences of two expressions involving square roots.
Rewrite the general term to see if the series is telescoping. Notice that each term looks like \( a_k = \frac{1}{\sqrt{k+2}} - \frac{1}{\sqrt{k}} \). Try to express the partial sums \( S_n = \sum_{k=1}^n a_k \) to identify cancellation patterns.
Write out the first few terms of the partial sum explicitly: \( S_n = \left( \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{1}} \right) + \left( \frac{1}{\sqrt{4}} - \frac{1}{\sqrt{2}} \right) + \cdots + \left( \frac{1}{\sqrt{n+2}} - \frac{1}{\sqrt{n}} \right) \). Group the positive and negative terms to observe which terms cancel out.
After cancellation, identify the remaining terms in \( S_n \). Typically, in telescoping series, most intermediate terms cancel, leaving only a few terms from the beginning and end of the sequence.
Analyze the limit of the partial sums \( S_n \) as \( n \to \infty \). If the limit exists and is finite, the series converges; otherwise, it diverges.

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Telescoping Series

A telescoping series is one where many terms cancel out when the series is expanded, simplifying the sum to a difference between a few terms. Recognizing this pattern helps in evaluating the limit of partial sums and determining convergence.
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Geometric Series

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding how to test for convergence, such as by examining partial sums or applying convergence tests, is essential to determine whether the series sums to a finite value.
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Convergence of an Infinite Series

Comparison and Limit Comparison Tests

These tests compare a given series to a known benchmark series to determine convergence or divergence. By comparing terms or their limits, one can infer the behavior of the original series, especially when direct evaluation is complex.
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Limit Comparison Test
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Domanda del libro di testo

43–44. Periodic doses

Suppose you take a dose of m mg of a particular medication once per day. Assume f equals the fraction of the medication that remains in your blood one day later. Just after taking another dose of medication on the second day, the amount of medication in your blood equals the sum of the second dose and the fraction of the first dose remaining in your blood, which is m + mf. Continuing in this fashion, the amount of medication in your blood just after your nth dose is


Aₙ = m + mf + ⋯ + mfⁿ⁻¹.


For the given values of f and m, calculate A₅, A₁₀, A₃₀, and lim (n → ∞) Aₙ. Interpret the meaning of the limit lim (n → ∞) Aₙ.


43.f = 0.25,m = 200 mg

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Domanda del libro di testo

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

∑ (from k = 1 to ∞) 2⁹k / kᵏ

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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 ∞) ((-7)ᵏ / k²)

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What test is advisable if a series involves a factorial term?

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Evaluate 1000!/998! without a calculator.

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45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 1 to ∞) (3/4)ᵏ

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