Skip to main content
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 10.7.31b
Chapter 10, Problem 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

Verified step by step guidance
1
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
Recommended video:
5:22
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.
Recommended video:
5:22
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.
Recommended video:
08:14
Using The Acceleration Function
Related Practice
Textbook Question

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.

1
views
Textbook Question

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.

Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


b.Find an explicit formula for the terms of the sequence.


Drug elimination

Jack took a 200-mg dose of a pain killer at midnight. Every hour, 5% of the drug is washed out of his bloodstream. Let dₙ be the amount of drug in Jack’s blood n hours after the drug was taken, where d₀ = 200mg.

1
views
Textbook Question

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).

{-5, 5, -5, 5, ......}

1
views
Textbook Question

71. Evaluating an infinite series two ways

Evaluate the series

∑ (k = 1 to ∞) (4 / 3ᵏ – 4 / 3ᵏ⁺¹) two ways.


b. Use a geometric series argument with Theorem 10.8.

1
views
Textbook Question

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


b. The sum ∑ (k = 3 to ∞) 1 / √(k − 2) is a p-series.