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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.1.133
Chapter 10, Problem 10.1.133

In Exercises 125–134, determine whether the sequence is monotonic, whether it is bounded, and whether it converges.
aₙ = (4ⁿ⁺¹ + 3ⁿ) / 4ⁿ

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Rewrite the given sequence \(a_n = \frac{4^{n+1} + 3^n}{4^n}\) by separating the terms in the numerator over the denominator: \(a_n = \frac{4^{n+1}}{4^n} + \frac{3^n}{4^n}\).
Simplify each term using the properties of exponents: \(\frac{4^{n+1}}{4^n} = 4^{(n+1)-n} = 4^1 = 4\), and \(\frac{3^n}{4^n} = \left(\frac{3}{4}\right)^n\).
Express the sequence in a simpler form: \(a_n = 4 + \left(\frac{3}{4}\right)^n\).
Analyze monotonicity by considering the term \(\left(\frac{3}{4}\right)^n\). Since \(\frac{3}{4} < 1\), this term decreases as \(n\) increases, so \(a_n\) is a decreasing sequence.
Determine boundedness and convergence: since \(\left(\frac{3}{4}\right)^n\) is always positive and approaches 0 as \(n \to \infty\), the sequence is bounded below by 4 and converges to 4.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Monotonic Sequences

A sequence is monotonic if it is either entirely non-increasing or non-decreasing. To determine monotonicity, compare consecutive terms or analyze the general term's behavior as n increases. Monotonic sequences simplify convergence analysis since they move consistently in one direction.
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Bounded Sequences

A sequence is bounded if all its terms lie within some fixed interval, meaning there exist real numbers M and m such that m ≤ aₙ ≤ M for all n. Boundedness is important because it restricts the sequence's values and is a key condition for convergence in monotonic sequences.
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Convergence of Sequences

A sequence converges if its terms approach a specific finite limit as n approaches infinity. To find the limit, simplify the general term and analyze dominant terms. Convergence indicates the sequence settles to a fixed value, which is crucial for understanding long-term behavior.
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Related Practice
Textbook Question

In Exercises 67–72, use the results of Exercises 63 and 64 to determine if each series converges or diverges.

∑(from n=2 to ∞) [(ln n)¹⁰⁰⁰ / n¹.⁰⁰¹]

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Textbook Question

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 = 0 to ∞) [ n xⁿ / (4ⁿ (n² + 1)) ]

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Textbook Question

Direct Comparison Test

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

∑ (from n=1 to ∞) (√n + 1) / (√(n² + 3))

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Textbook Question

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑ (from n = 2 to ∞) [(ln n / n)³]

Textbook Question

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 ∞) sin (1/n)

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Textbook Question

Applying the Integral Test

Use the Integral Test to determine if the series in Exercises 1–12 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

∑ (from n = 2 to ∞) ln(n²) / n

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