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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 10.7.31b
Capitolo 10, Problema 10.7.31b

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


b. (2n)! / (2n − 1)! = 2n

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Recall the definition of factorial: for any positive integer \(k\), \(k! = k \times (k-1) \times (k-2) \times \cdots \times 1\).
Write out the factorial expressions explicitly: \((2n)! = (2n) \times (2n - 1) \times (2n - 2) \times \cdots \times 1\) and \((2n - 1)! = (2n - 1) \times (2n - 2) \times \cdots \times 1\).
Set up the fraction and simplify by canceling common terms: \(\frac{(2n)!}{(2n - 1)!} = \frac{(2n) \times (2n - 1) \times \cdots \times 1}{(2n - 1) \times \cdots \times 1} = 2n\).
Conclude that the statement \(\frac{(2n)!}{(2n - 1)!} = 2n\) is true because the factorial terms cancel except for the factor \$2n$ in the numerator.
Therefore, the expression simplifies exactly to \$2n$, confirming the statement is correct.

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Factorials and Their Properties

A factorial, denoted n!, is the product of all positive integers from 1 up to n. Understanding how factorials expand and simplify is essential, especially when dealing with ratios like (2n)! / (2n − 1)!, where many terms cancel out.
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Factorials

Simplifying Factorial Expressions

When dividing factorials such as (2n)! by (2n − 1)!, most terms cancel, leaving only the highest term, 2n. This simplification helps determine if the given expression equals 2n or not.
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Factorials

Verification Using Counterexamples

To test the truth of a statement involving factorials, substituting specific values of n can confirm or refute it. A single counterexample disproves the statement, while consistent results support its validity.
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Using The Acceleration Function
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Domanda del libro di testo

Explain why or why not

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


b.If a sequence of positive numbers converges, then the sequence is decreasing.

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67–70. Formulas for sequences of partial sums Consider the following infinite series.


a.Find the first four partial sums S₁, S₂, S₃, S₄ of the series.


∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

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57–60. Heights of bouncing balls A ball is thrown upward to a height of hₒ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hₙ be the height after the nth bounce. Consider the following values of hₒ and r.


b. Find an explicit formula for the nth term of the sequence {hₙ}.


h₀ = 20,r = 0.5

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87. Explain why or why not

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


b. If ∑ (k = 1 to ∞) aₖ diverges, then ∑ (k = 10 to ∞) aₖ diverges.

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18–20. Evaluating geometric series two ways Evaluate each geometric series two ways.


b. Evaluate the series using Theorem 10.7.


∑ (k = 0 to ∞) (–2/7)ᵏ

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39–40. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, use Theorem 10.13 to complete the following.


b. Find an upper bound for the remainder Rₙ.


39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2

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