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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 10.4.11
Capitolo 10, Problema 10.4.11

Limit Comparison Test
In Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.
∑ (from n=1 to ∞) n(n + 1) / ((n² + 1)(n − 1))

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Identify the given series: \( \sum_{n=1}^{\infty} \frac{n(n + 1)}{(n^2 + 1)(n - 1)} \). We want to determine if it converges or diverges using the Limit Comparison Test.
Choose a comparison series \( b_n \) that resembles the behavior of \( a_n = \frac{n(n + 1)}{(n^2 + 1)(n - 1)} \) for large \( n \). Simplify the dominant terms: numerator behaves like \( n^2 \), denominator behaves like \( n^3 \), so \( a_n \) behaves like \( \frac{n^2}{n^3} = \frac{1}{n} \). Thus, choose \( b_n = \frac{1}{n} \).
Compute the limit \( L = \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\frac{n(n + 1)}{(n^2 + 1)(n - 1)}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{n(n + 1) \cdot n}{(n^2 + 1)(n - 1)} \).
Simplify the expression inside the limit and evaluate \( L \). If \( L \) is a finite positive number (i.e., \( 0 < L < \infty \)), then \( a_n \) and \( b_n \) have the same behavior regarding convergence.
Since \( \sum \frac{1}{n} \) (the harmonic series) diverges, use the result of the Limit Comparison Test to conclude whether the original series converges or diverges.

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Limit Comparison Test

The Limit Comparison Test is used to determine the convergence or divergence of an infinite series by comparing it to a second series with known behavior. It involves taking the limit of the ratio of the nth terms of the two series. If the limit is a positive finite number, both series either converge or diverge together.
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Limit Comparison Test

Behavior of Rational Functions for Large n

When analyzing series with rational expressions, it is important to understand the dominant terms as n approaches infinity. Simplifying the highest degree terms in numerator and denominator helps approximate the general term's behavior, which is crucial for choosing an appropriate comparison series.
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Intro to Rational Functions

Convergence of p-Series

A p-series is a series of the form ∑ 1/n^p, which converges if p > 1 and diverges otherwise. Recognizing or approximating a given series to a p-series helps in applying the Limit Comparison Test effectively, as p-series have well-known convergence properties.
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Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?

∑ (from n = 1 to ∞) [ (x − 1)ⁿ / √n ]

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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 ∞) (n / (3n + 1))ⁿ

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Find the first four nonzero terms of the Taylor series for the functions in Exercises 1–10.

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If ∑aₙ is a convergent series of positive terms, prove that ∑sin(aₙ) converges.

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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 ∞) (tanh n) / n²

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Direct Comparison Test

In Exercises 1–8, use the Direct Comparison Test to determine if each series converges or diverges.

∑ (from n=1 to ∞) cos²n / n^(3/2)

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