Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.2.43

Chapter 10, Problem 10.2.43

13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.

{√((1 + 1 / 2n)ⁿ)}

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Identify the given sequence: \(a_n = \sqrt{\left(1 + \frac{1}{2n}\right)^n}\).

Rewrite the sequence to a form that is easier to analyze by expressing the square root as a power: \(a_n = \left(1 + \frac{1}{2n}\right)^{\frac{n}{2}}\).

Recognize that the expression inside the parentheses resembles the form \(\left(1 + \frac{1}{m}\right)^m\) which is related to the number \(e\) as \(m \to \infty\).

Set \(m = 2n\) so that the expression becomes \(\left(1 + \frac{1}{m}\right)^{\frac{m}{2}}\) and analyze the limit as \(m \to \infty\).

Use the known limit \(\lim_{m \to \infty} \left(1 + \frac{1}{m}\right)^m = e\) to conclude that \(\lim_{n \to \infty} a_n = e^{\frac{1}{2}}\).

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

Exponential and Root Functions in Limits

When sequences involve expressions like powers and roots, it is important to simplify or rewrite them using properties of exponents and radicals. This often helps in identifying the behavior of the sequence as the index grows large.

Limit of (1 + 1/n)^n as n Approaches Infinity

The sequence (1 + 1/n)^n is a classic limit that approaches the mathematical constant e (~2.718). Recognizing this limit helps in evaluating more complex sequences that include similar expressions raised to powers.

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Textbook Question

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.

∑ (k = 1 to ∞) k / eᵏ

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Textbook Question

39–44. {Use of Tech} Estimating infinite series Estimate the value of the following convergent series with an absolute error less than 10⁻³.

∑ (k = 1 to ∞) (−1)ᵏ / kᵏ

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Textbook Question

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

∑ (k = 1 to ∞) (−1)ᵏ⁺¹ / (2√k − 1)

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Textbook Question

33–38. {Use of Tech} Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude. Although you do not need it, the exact value of the series is given in each case.

π / 4 = ∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

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Textbook Question

55–70. More sequences

Find the limit of the following sequences or determine that the sequence diverges.

{(75n⁻¹ / 99ⁿ) + (5ⁿsinn / 8ⁿ)}

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Textbook Question

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

∑ (k = 1 to ∞) 1 / (2k − √k)

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13–52. Limits of sequencesFind the limit of the following sequences or determine that the sequence diverges.{√((1 + 1 / 2n)ⁿ)}

Verified video answer for a similar problem:

Key Concepts

Limits of Sequences

Exponential and Root Functions in Limits

Limit of (1 + 1/n)^n as n Approaches Infinity

13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.

{√((1 + 1 / 2n)ⁿ)}