Question

Determine the values of $n \in \mathbb{Z}$ for which the inequality $2^n < n!$. Write down a proposition containing your result and prove it. (Hint: Note that $(n+1)! = (n+1) \cdot n!$. Also, recall that $0! = 1$ and $!$ is undefined for negative integers.)

Name: determine the values of n in mathbbz for which the inequality 2n n write down a proposition containing your result and prove it hint note that n1 n1 cdot n also recall that 0 1 and is undefi 61292
Uploaded: 2023-03-15T03:16:22-08:00
Duration: 2 min 26 s
Channel: Nick Johnson
Description: determine the values of n in mathbbz for which the inequality 2n n write down a proposition containing your result and prove it hint note that n1 n1 cdot n also recall that 0 1 and is undefi 61292

          Determine the values of $n \in \mathbb{Z}$ for which the inequality $2^n < n!$.
Write down a proposition containing your result and prove it.
(Hint: Note that $(n+1)! = (n+1) \cdot n!$. Also, recall that $0! = 1$ and $!$ is undefined for negative integers.)

determine the values of n in mathbbz for which the inequality 2n n write down a proposition containing your result and prove it hint note that n1 n1 cdot n also recall that 0 1 and is undefi 61292

Added by Dana B.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Nick Johnson

03/15/2023

Step 1

For $n=1$, $2^1 = 2$ and $1! = 1$, so $2^1 > 1!$. For $n=2$, $2^2 = 4$ and $2! = 2$, so $2^2 > 2!$. For $n=3$, $2^3 = 8$ and $3! = 6$, so $2^3 > 3!$. For $n=4$, $2^4 = 16$ and $4! = 24$, so $2^4 < 4!$. For $n=5$, $2^5 = 32$ and $5! = 120$, so $2^5 < 5!$. For Show more…

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Determine the values of $n \in \mathbb{Z}$ for which the inequality $2^n < n!$. Write down a proposition containing your result and prove it. (Hint: Note that $(n+1)! = (n+1) \cdot n!$. Also, recall that $0! = 1$ and $!$ is undefined for negative integers.)