Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 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

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)!.

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.

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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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

Explain why or why not

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

a. If the Limit Comparison Test can be applied successfully to a given series with a certain comparison series, the Comparison Test also works with the same comparison series.

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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

{Use of Tech} Drug Dosing

A patient takes 75 mg of a medication every 12 hours; 60% of the medication in the blood is eliminated every 12 hours.

a.Let dₙ equal the amount of medication (in mg) in the bloodstream after n doses, where d₁ = 75.

Find a recurrence relation for dₙ.

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