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November 3rd, 2019, 07:28 AM   #1
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Range of X

Find the range of X such that $\displaystyle (X+6)^{n} \cdot n!\geq n^n \; , n\in \mathbb{N} ,X\in \mathbb{R}.$
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November 3rd, 2019, 07:59 AM   #2
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unfortunately there's no definitive answer to this question as you have two variables rather than 1.

setting n=2 will give a different range for x than setting it to 9.
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November 3rd, 2019, 08:32 AM   #3
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Quote:
Originally Posted by phillip1882 View Post
unfortunately there's no definitive answer to this question as you have two variables rather than 1.

setting n=2 will give a different range for x than setting it to 9.
he's looking for the infinite intersection of all ranges of X over all the natural numbers.
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November 3rd, 2019, 09:48 AM   #4
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Originally Posted by idontknow View Post
Find the range of X such that $\displaystyle (X+6)^{n} \cdot n!\geq n^n \; , n\in \mathbb{N} ,X\in \mathbb{R}.$
$$\lim\limits_{n \rightarrow \infty} \dfrac{n}{(n!)^{1/n}} = e$$
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