The video explains the limit of the expression (n! / n^n) as n approaches infinity. Initially, it poses the question of where this limit might land, suggesting possibilities like infinity, a specific number, or zero. The presenter then breaks down the components: n factorial (n!) and n to the power of n (n^n). By examining the first 'n' terms, it's shown that n! is the product of integers from 1 to n, while n^n is n multiplied by itself 'n' times. The core of the explanation lies in comparing the terms of n! to n^n. For each pair of corresponding terms in the expansion of n! and n^n, the numerator term (from n!) is less than or equal to the denominator term (from n^n). This leads to the conclusion that each factor in the ratio (n! / n^n) is at most 1. Specifically, the first factor is 1/n. The presenter emphasizes that the entire product cannot be larger than its first factor, which is 1/n. To formally prove the limit is zero, the squeeze theorem is invoked. Since the limit of 0 is 0 and the limit of 1/n as n approaches infinity is also 0, the limit of the expression in question is 'squeezed' to 0. For those interested in the rate at which the expression approaches zero, the video mentions Stirling's approximation, which simplifies the behavior to a geometric progression of 1/e^n. The video concludes by noting that while n! grows very rapidly, n^n grows even faster, causing the ratio to tend towards zero.

BriTheMathGuy

Take a look at this limit. The top explodes and the bottom explodes. So, where do you think it lands? Infinity, a number, maybe this ties?

Well, it's going to crash straight to zero. You can figure it out by writing out the first n terms. n factorial goes 1, 2, 3 all the way up to n. Where n to the power of n just means n * n * n n times. For every one of those pairs, each numerator is no bigger than its denominator pair n. And every one of those factors is at most one. The very first factor being 1 over n. This is a fancy way of saying the entire product can't be larger than that first factor. You can check it with yourself for a few examples of n.

And so now all you need is the squeeze theorem. The limit of zero tends to zero, the limit of 1 over n tends to zero, and so our limit in question is squeezed in the middle, tends to zero. If you want to know how fast it crashes towards zero, you can use Stirling's approximation here. Essentially, this turns into a geometric progression of 1 over e to the n. That's about how fast it tends towards zero. n factorial grows fast enough to beat almost anything. It's just n to the n isn't almost anything.

This Limit Crashes Straight to Zero

Take a look at this limit. The top
explodes and the bottom explodes.
So, where do you think it lands?
Infinity, a number, maybe this ties?
Well, it's going to crash straight to
zero.
You can figure it out by writing out the
first n terms.
n factorial goes 1 2 3 all the way up to
n.
Where n to the power of n just means n *
n * n n times.
For every one of those pairs, each
numerator is no bigger than its
denominator pair n.
And every one of those factors is at
most one.
The very first factor being 1 over n.
This is a fancy way of saying the entire
product can't be larger than that first
factor.
You can check it with yourself for a few
examples of n.
And so now all you need is the squeeze
theorem.
The limit of zero tends to zero, the
limit of 1 over n tends to zero, and so
our limit in question is squeezed in the
middle, tends to zero.
If you want to know how fast it crashes
towards zero, you can use Stirling's
approximation here.
Essentially, this turns into a geometric
progression of 1 over e to the n. That's
about how fast it tends towards zero.
n factorial grows fast enough to beat
almost anything. It's just n to the n
isn't almost anything.

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This Limit Crashes Straight to Zero

Transcription