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
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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)}
Verified step by step guidance1
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.
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8:22
Introduction to Sequences
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.
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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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Related Practice
Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(n + 1)!⁄n!}
Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
31.∑ (k = 1 to ∞) 2^(–3k)
Textbook Question
49–50. Limits from graphs Consider the following sequences. Find the first four terms of the sequence .Based on part (a) and the figure, determine a plausible limit of the sequence.
aₙ = 2 + 2⁻ⁿ;n = 1, 2, 3, …
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Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) k⁴ / (eᵏ⁵)
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