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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblema 10.4.17
Capitolo 10, Problema 10.4.17

Determining Convergence or Divergence
Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.
∑ (from n=1 to ∞) 1 / (2√n + ³√n)

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Identify the general term of the series: \(a_n = \frac{1}{2\sqrt{n} + \sqrt[3]{n}}\).
Analyze the behavior of the denominator for large \(n\). Since \(\sqrt{n} = n^{1/2}\) and \(\sqrt[3]{n} = n^{1/3}\), the term \(2\sqrt{n}\) grows faster than \(\sqrt[3]{n}\) as \(n\) increases.
Approximate the general term for large \(n\) by focusing on the dominant term in the denominator: \(a_n \approx \frac{1}{2n^{1/2}}\).
Compare the series to a known benchmark series, such as the \(p\)-series \(\sum \frac{1}{n^p}\), where convergence depends on \(p\). Here, the comparison is with \(\sum \frac{1}{n^{1/2}}\).
Recall that the \(p\)-series \(\sum \frac{1}{n^p}\) converges if and only if \(p > 1\). Since \(1/2 < 1\), the comparison series diverges, so by the Comparison Test, the original series also diverges.

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Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining whether a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. If the sum does not approach a finite value, the series diverges.
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Convergence of an Infinite Series

Comparison Test

The Comparison Test helps determine convergence by comparing the given series to a known benchmark series. 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. This test is useful when terms are positive and can be bounded.
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Direct Comparison Test

Asymptotic Behavior of Terms

Analyzing the dominant behavior of terms for large n helps simplify complex expressions. For example, in 1/(2√n + ³√n), the term with the slower decay rate (smaller exponent) dominates. Understanding which term controls the behavior is key to applying convergence tests effectively.
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Asymptotes of Hyperbolas
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Error Estimates

The approximation eˣ = 1 + x + (x² / 2) is used when x is small. Use the Remainder Estimation Theorem to estimate the error when |x| < 0.1.

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Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = 2 + (0.1)ⁿ

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Domanda del libro di testo

Convergence or Divergence

Which of the series in Exercises 57–64 converge, and which diverge? Give reasons for your answers.

∑ (from n = 1 to ∞) [(-1)ⁿ (n!)ⁿ] / [n^(n²)]

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Domanda del libro di testo

In Exercises 53–56, determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001.

∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (n / (n² + 1))]

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Domanda del libro di testo

Using the Ratio Test

In Exercises 1–8, use the Ratio Test to determine whether each series converges absolutely or diverges.

∑(from n=2 to ∞) [(3ⁿ⁺²) / ln(n)]

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Domanda del libro di testo

Make up a geometric series ∑a rⁿ⁻¹ that converges to the number 5 if

b. a = 13/2

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