Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.R.19

Chapter 10, Problem 10.R.19

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = ((3n² + 2n + 1) · sin(n)) / (4n³ + n) (Hint: Use the Squeeze Theorem.)

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Identify the given sequence: $a_n = \frac{(3n^2 + 2n + 1) \cdot \sin(n)}{4n^3 + n}$.

Analyze the behavior of the numerator and denominator separately as $n$ approaches infinity. The numerator grows roughly like \$3n^2$ times \(\sin(n)$, and the denominator grows like \)4n^3$.

Note that $\sin(n)$ oscillates between $-1$ and $1$, so $-1 \leq \sin(n) \leq 1$. Use this to create inequalities for $a_n$:

\[-\frac{3n^2 + 2n + 1}{4n^3 + n} \leq a_n \leq \frac{3n^2 + 2n + 1}{4n^3 + n}.\]

Simplify the bounding expressions by dividing numerator and denominator by $n^3$ to understand their limits:

\[-\frac{3/n + 2/n^2 + 1/n^3}{4 + 1/n^2} \leq a_n \leq \frac{3/n + 2/n^2 + 1/n^3}{4 + 1/n^2}.\]

Apply the Squeeze Theorem: since both bounds approach $0$ as $n \to \infty$, conclude that $\lim_{n \to \infty} a_n = 0$.

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Key Concepts

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

Limits of Sequences

The limit of a sequence describes the value that the terms of the sequence approach as the index n goes to infinity. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.

Squeeze Theorem

The Squeeze Theorem helps find limits by 'trapping' a sequence between two others that have the same limit. If aₙ ≤ bₙ ≤ cₙ for all n beyond some point, and both aₙ and cₙ converge to L, then bₙ also converges to L.

Behavior of Trigonometric Functions in Limits

Trigonometric functions like sin(n) oscillate between -1 and 1 and do not have limits as n approaches infinity. When combined with sequences that tend to zero, their bounded nature allows the use of the Squeeze Theorem to evaluate the overall limit.

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Related Practice

Textbook Question

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = (–1)ⁿ / 0.9ⁿ

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Textbook Question

27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.

∑ (from k = 1 to ∞)2ᵏ / 3ᵏ⁺²

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

c. The terms of the sequence of partial sums of the series ∑ aₖ approach 5/2, so the infinite series converges to 5/2.

Textbook Question

Give an example (if possible) of a sequence {aₖ} that converges, while the series ∑ (from k = 1 to ∞) aₖ diverges.

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Textbook Question

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

e.The sequence aₙ = n² / (n² + 1) converge.

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Textbook Question

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞)k√k / k³

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.aₙ = ((3n² + 2n + 1) · sin(n)) / (4n³ + n) (Hint: Use the Squeeze Theorem.)

Verified video answer for a similar problem:

Key Concepts

Limits of Sequences

Squeeze Theorem

Behavior of Trigonometric Functions in Limits

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = ((3n² + 2n + 1) · sin(n)) / (4n³ + n) (Hint: Use the Squeeze Theorem.)