Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
35.∑ (k = 0 to ∞) 3(–π)^(–k)
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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 10.2.37
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{√(n² + 1) − n}
Verified step by step guidance1
Start by examining the sequence given: \(a_n = \sqrt{n^2 + 1} - n\).
To find the limit as \(n\) approaches infinity, consider rationalizing the expression by multiplying and dividing by the conjugate: \(\sqrt{n^2 + 1} + n\).
Rewrite the sequence as \(a_n = \frac{(\sqrt{n^2 + 1} - n)(\sqrt{n^2 + 1} + n)}{\sqrt{n^2 + 1} + n} = \frac{n^2 + 1 - n^2}{\sqrt{n^2 + 1} + n} = \frac{1}{\sqrt{n^2 + 1} + n}\).
Analyze the denominator \(\sqrt{n^2 + 1} + n\) as \(n\) becomes very large. Since \(\sqrt{n^2 + 1}\) behaves like \(n\) for large \(n\), the denominator grows approximately like \$2n$.
Therefore, the sequence behaves like \(\frac{1}{2n}\) for large \(n\), and you can conclude the limit by considering the behavior of this simplified form as \(n\) approaches infinity.
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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 becomes very large. 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
Algebraic Manipulation for Limits
When evaluating limits involving expressions like √(n² + 1) − n, algebraic techniques such as rationalizing the expression by multiplying numerator and denominator by the conjugate help simplify the expression and reveal the limit more clearly.
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Behavior of Functions as n Approaches Infinity
Understanding how functions behave as n grows large, especially dominant terms like n² inside a square root, is crucial. Recognizing that √(n² + 1) behaves like n for large n helps in approximating and finding the limit.
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Related Practice
Textbook Question
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 2 to ∞) (−1)ᵏ · k · (k² + 1) / (k³ − 1)
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Textbook Question
35–44. Limits of sequences Write the terms a₁, a₂, a₃, and a₄ of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why.
{Use of Tech} aₙ₊₁ = (aₙ⁄₁₁ )+ 50;a₀ = 50
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Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(√(4n⁴ + 3n))⁄(8n² + 1)}
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Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{2ⁿ⁺¹3⁻ⁿ}
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Textbook Question
Find two different explicit formulas for the sequence {1, -2, 3, -4, -5 .....}
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