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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.2.37
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 guidance
1
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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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

The first ten terms of the sequence {(1 + 1/10ⁿ)^10ⁿ}∞ ₙ₌₁ are rounded to 8 digits right of the decimal point (see table). Make a conjecture about the limit of the sequence.

n an

1 2.59374246

2 2.70481383

3 2.71692393

4 2.71814593

5 2.71826824

6 2.71828047

7 2.71828169

8 2.71828179

9 2.71828204

10 2.71828203

1
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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.


65. ∑ (k = 1 to ∞) (1 / √(k + 1) – 1 / √(k + 3))

Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from j = 2 to ∞)1 / (j ln¹⁰j)

Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞)cos(1 / k⁹)

Textbook Question

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

Textbook Question

13–52. Limits of sequences

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


{2ⁿ⁺¹3⁻ⁿ}