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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 10.7.31a
Chapter 10, Problem 10.7.31a

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.

Verified step by step guidance
1
Recall the definition of the factorial function: for a positive integer \(n\), \(n! = 1 \times 2 \times 3 \times \cdots \times n\).
Understand the expressions given: \(n!n!\) means multiplying \(n!\) by itself, while \((2n)!\) means the factorial of \$2n$, which is \(1 \times 2 \times 3 \times \cdots \times (2n)\).
To check if \(n!n! = (2n)!\) for all positive integers \(n\), consider testing small values of \(n\) to see if the equality holds.
For example, when \(n=1\), \(n!n! = 1! \times 1! = 1 \times 1 = 1\) and \((2n)! = (2 \times 1)! = 2! = 2\), so the equality does not hold here.
Since the equality fails for \(n=1\), the statement is false for all positive integers \(n\). This counterexample disproves the statement.

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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 to n. Understanding how factorials grow and their basic properties is essential to compare expressions like n!·n! and (2n)!.
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Factorials

Inequalities and Growth Rates of Factorials

Factorials grow very rapidly, and (2n)! grows faster than n!·n!. Recognizing this difference helps determine whether n!·n! equals (2n)! or not by comparing their magnitudes.
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Factorials

Counterexamples in Mathematical Proof

To disprove a statement, providing a single counterexample suffices. Testing the equality for small values of n can quickly show whether the statement holds universally or not.
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Slopes of Tangent Lines
Related Practice
Textbook Question

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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Textbook Question

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.

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Textbook Question

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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Textbook Question

Express 0.314141414… as a ratio of two integers.

Textbook Question

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.


a. Find an upper bound for the remainder in terms of n.


41. ∑ (k = 1 to ∞) 1 / k⁶

Textbook Question

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?