Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)(7ᵏ + 11ᵏ) / 11ᵏ
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 10.2.47
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(1 / n)¹⁄ⁿ}
Verified step by step guidance1
Identify the given sequence: \(a_n = \left( \frac{1}{n} \right)^{\frac{1}{n}}\).
Rewrite the sequence using properties of exponents and logarithms to make the limit easier to analyze: \(a_n = e^{\ln \left( \left( \frac{1}{n} \right)^{\frac{1}{n}} \right)} = e^{\frac{1}{n} \ln \left( \frac{1}{n} \right)}\).
Simplify the exponent: \(\frac{1}{n} \ln \left( \frac{1}{n} \right) = \frac{1}{n} (-\ln n) = -\frac{\ln n}{n}\).
Analyze the limit of the exponent as \(n\) approaches infinity: \(\lim_{n \to \infty} -\frac{\ln n}{n}\). Recall that \(\ln n\) grows slower than \(n\), so this limit tends to zero.
Conclude the limit of the sequence by applying the limit to the exponential form: \(\lim_{n \to \infty} a_n = e^{\lim_{n \to \infty} -\frac{\ln n}{n}} = e^0\).
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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 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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8:22
Introduction to Sequences
Properties of Exponents and Roots
Understanding how to manipulate expressions with fractional exponents and roots is essential. For example, (1/n)^(1/n) can be rewritten using exponent rules to analyze its behavior as n increases.
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06:21Properties of Functions
Use of Logarithms in Limit Evaluation
Taking the natural logarithm of a sequence can simplify limit evaluation, especially for expressions involving exponents. By analyzing the limit of the logarithm, one can often find the original sequence's limit using continuity of the exponential function.
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Related Practice
Textbook Question
54–69. Telescoping series
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.
67. ∑ (k = 1 to ∞) 3 / (k² + 5k + 4)
Textbook Question
6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.
{1.00001ⁿ}
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Textbook Question
Compare the growth rates of {n¹⁰⁰} and {eⁿ⁄¹⁰⁰} as n → ∞.
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