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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.2.83a
Capitolo 10, Problema 10.2.83a

Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
a.If limₙ→∞aₙ = 1 and limₙ→∞bₙ = 3, then limₙ→∞(bₙ / aₙ) = 3.

Guida verificata passo dopo passo
1
Recall the limit laws for sequences: if \( \lim_{n \to \infty} a_n = A \) and \( \lim_{n \to \infty} b_n = B \), and if \( A \neq 0 \), then \( \lim_{n \to \infty} \frac{b_n}{a_n} = \frac{B}{A} \).
Given \( \lim_{n \to \infty} a_n = 1 \) and \( \lim_{n \to \infty} b_n = 3 \), since \( a_n \) approaches 1 (which is not zero), the limit of the quotient should be \( \frac{3}{1} = 3 \).
Therefore, the statement \( \lim_{n \to \infty} \frac{b_n}{a_n} = 3 \) is true under these conditions.
To confirm, consider that the denominator sequence \( a_n \) does not approach zero, so division by \( a_n \) is well-defined in the limit.
Hence, the limit of the quotient is the quotient of the limits, which justifies the statement.

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Concetti chiave

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Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the index goes to infinity. If a sequence converges to a limit, its terms get arbitrarily close to that limit for sufficiently large indices.
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Limit Laws for Sequences

Limit laws allow us to compute limits of combined sequences using operations like addition, multiplication, and division, provided the limits of individual sequences exist and, in the case of division, the denominator's limit is not zero.
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Counterexamples in Limit Problems

Counterexamples demonstrate that a general statement is false by providing a specific case where the statement fails. In limit problems, they help test the validity of limit laws or assumptions, especially when conditions like nonzero denominators are involved.
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88–89. Binary numbers

Humans use the ten digits 0 through 9 to form base-10 or decimal numbers, whereas computers calculate and store numbers internally as binary numbers—numbers consisting entirely of 0’s and 1’s. For this exercise, we consider binary numbers that have the form 0.b₁b₂b₃⋯, where each of the digits b₁, b₂, b₃, ⋯ is either 0 or 1. The base-10 representation of the binary number 0.b₁b₂b₃⋯ is the infinite series

b₁ / 2¹ + b₂ / 2² + b₃ / 2³ + ⋯


89. Computers can store only a finite number of digits and therefore numbers with nonterminating digits must be rounded or truncated before they can be used and stored by a computer.


a. Find the base-10 representation of the binary number 0.001̅1.

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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. The sum ∑ (k = 1 to ∞) 1 / 3ᵏ is a p-series.

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87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


a. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 10 to ∞) aₖ converges.

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Loglog p-series Consider the series ∑ (k = 2 to ∞) 1 / (k(ln k)(ln ln k)ᵖ), where p is a real number.

a. For what values of p does this series converge?

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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. n!n! = (2n)! for all positive integers n.

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{Use of Tech} Repeated square roots

Consider the sequence defined by

aₙ₊₁ = √(2 + aₙ),a₀ = √2, for n = 0, 1, 2, 3, …


a.Evaluate the first four terms of the sequence {aₙ}.

State the exact values first, and then the approximate values.

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