Show that log n! is greater than (n log n)/4 for n > 4.

Chapter 2, Problem 73E

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QUESTION:

Show that \(\log n !\) is greater than \((n \log n) / 4\) for \(n>4\). [Hint: Begin with the inequality \(n !>n(n-1)(n-2) \cdots n / 2 .]\)

Let \(f(x)\) and \(g(x)\) be functions from the set of real numbers to the set of real numbers. We say that the functions \(f\) and \(g\) are asymptotic and write \(f(x) \sim g(x)\) if \(\lim _{x \rightarrow \infty} f(x) / g(x)=1\)

Equation Transcription:

 · · · .]

Text Transcription:

(n log n)/4

n > 4

n! > n(n − 1)(n − 2) dot n/2.]

f(x)

g(x)

f

g

f (x) ∼ g(x)

Lim_x right arrow infinity  f (x)/g(x) = 1

Questions & Answers

QUESTION:

Show that \(\log n !\) is greater than \((n \log n) / 4\) for \(n>4\). [Hint: Begin with the inequality \(n !>n(n-1)(n-2) \cdots n / 2 .]\)

Let \(f(x)\) and \(g(x)\) be functions from the set of real numbers to the set of real numbers. We say that the functions \(f\) and \(g\) are asymptotic and write \(f(x) \sim g(x)\) if \(\lim _{x \rightarrow \infty} f(x) / g(x)=1\)

Equation Transcription:

 · · · .]

Text Transcription:

(n log n)/4

n > 4

n! > n(n − 1)(n − 2) dot n/2.]

f(x)

g(x)

f

g

f (x) ∼ g(x)

Lim_x right arrow infinity  f (x)/g(x) = 1

ANSWER:

SOLUTION

Step 1

In this problem, we have to show that  for .

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