Textbook Question
17–22. Integral Test Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied.
∑ (k = 1 to ∞) 1 / (∛(5k + 3))
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 10.2.49
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{bₙ}, where
bₙ = { n / (n + 1)if n ≤ 5000
ne⁻ⁿif n > 5000 }
Verified step by step guidance1
Understand the definition of the sequence \( b_n \): it has two different expressions depending on whether \( n \) is less than or equal to 5000 or greater than 5000.
Analyze the first part of the sequence for \( n \leq 5000 \): \( b_n = \frac{n}{n+1} \). Since this is a rational function, consider the limit as \( n \to \infty \) to understand its behavior.
Analyze the second part of the sequence for \( n > 5000 \): \( b_n = n e^{-n} \). Recognize that \( e^{-n} = \frac{1}{e^n} \) and consider the limit as \( n \to \infty \) to determine how this term behaves.
Since the sequence changes definition at \( n = 5000 \), check the limit of both parts as \( n \to \infty \) to see if they approach the same value or if the sequence diverges.
Conclude the overall limit of the sequence \( b_n \) by combining the results from both parts and considering the behavior for large \( n \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit of a Sequence
The limit of a sequence is 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.
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Introduction to Sequences
Piecewise-Defined Sequences
A piecewise-defined sequence has different formulas for different ranges of n. To find its limit, analyze the behavior of each piece separately, especially the part that applies as n approaches infinity.
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Exponential Decay and Its Limit
Sequences involving terms like n·e⁻ⁿ combine polynomial growth with exponential decay. Since exponential decay dominates polynomial growth, such terms tend to zero as n approaches infinity.
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Related Practice
Textbook Question
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = 1/10ⁿ
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Textbook Question
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k¹/ᵏ
Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
29.∑ (k = 1 to ∞) e^(–2k)
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Textbook Question
9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.
∑ (k = 0 to ∞) 1 / (1000 + k)
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Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
33.∑ (k = 4 to ∞) 1 / 5ᵏ
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