For which nonnegative integers n is. n^2\leq n!?

midtlinjeg 2021-09-10 Answered
For which nonnegative integers n is
n2n!?
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Expert Answer

Nathaniel Kramer
Answered 2021-09-11 Author has 78 answers

Let us determine the possible integers for which n2n! by constructing a graph of the two functions f(x)=x2 and g(x)=x!
The intersection of the two graphs is then (3.5624,12.6906). For values n larger than 3.5624, the inequality n2n! will then hold.
Since n is an integer:
n>3
Moreover, we note that the equality holds for n=0andn=1 as well, since 010=02 and 11=12

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