Textbook Question
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = (2ⁿ⁺¹) / (2ⁿ + 1)
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14. Sequences & Series
Sequences
2:16 minutesProblem 10.1.37
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.
aₙ = 1⁄10ⁿ; n = 1, 2, 3, …
Verified step by step guidance1
Identify the general term of the sequence given by \(a_{n} = \frac{1}{10^{n}}\) for \(n = 1, 2, 3, \ldots\).
Calculate the first four terms by substituting \(n = 1, 2, 3, 4\) into the formula: \(a_{1} = \frac{1}{10^{1}}\), \(a_{2} = \frac{1}{10^{2}}\), \(a_{3} = \frac{1}{10^{3}}\), and \(a_{4} = \frac{1}{10^{4}}\).
Observe the pattern of the terms: each term is one-tenth of the previous term, so the terms get smaller as \(n\) increases.
To determine if the sequence converges, consider the limit of \(a_{n}\) as \(n\) approaches infinity: \(\lim_{n \to \infty} \frac{1}{10^{n}}\).
Since \$10^{n}\( grows without bound as \)n$ increases, the terms approach zero, so the sequence converges and its limit is 0.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sequences and Their Terms
A sequence is an ordered list of numbers defined by a specific formula for its nth term. Understanding how to compute the first few terms by substituting values of n helps identify the pattern and behavior of the sequence.
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Limit of a Sequence
The limit of a sequence is the value that the terms approach as n becomes very large. If the terms get arbitrarily close to a fixed number, the sequence converges to that limit; otherwise, it diverges.
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Convergence and Divergence
A sequence converges if its terms approach a finite number as n increases indefinitely. If the terms do not approach any finite value or grow without bound, the sequence diverges. Recognizing this helps in making conjectures about the sequence's behavior.
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