Limit of n!/n^n as n goes to infinity, squeeze theorem, calculus 2 tutorial

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limit of n!/n^n as n goes to infinity, plus the list, and squeeze theorem

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When we went over this in my calc 1 class, my professor called this the oreo theorem, and then handed out oreos for the entire class. Seems appropriate!

matthewgiallourakis

i'm preparing for my finals completely with the help of your videos and i really really appreciate the help. you're a genius. thank you for your work!

zaerrr

You are a great teacher with a bubbly personality!

kujayasinghe

I hope you explain the gamma function soon.
Remember learning it at university but never really understood it.
I strongly believe that you can teach it.

Really enjoy your videos :)

Botisaurus

Dude, just awesome! I felt like a kid again watching your video.... :) Something about the whole 'math for the sake of math' vibe you had going. Please keep up the interesting videos. -Float Circuit.

ozzyfromspace

I am really appreciated I couldnt find any explanational video like this on youtube nice job bro

ulasaltunn

Great video. Loved your way of doing this. What i did was express the factorial with the gamma function and then differentiated it. Then some trivial work was needed, but i arrived at the same result.

aubertducharmont

Dude your videos are really awesome and motivating. Thanks man!

yajurphullera

I would have used a similar approach initially, expanding (n!/nⁿ) as (1 · 2 · 3 · 4 · ... · n) / (n · n · n · n · ... · n), and then observing how (nⁿ) grows faster than (n!), since there are more linear terms in the expansion of (nⁿ) than (n!), when n approaches ∞...

The main difference to my approach, however, would be applying one of these following rules for limits of quotients between two functions:
· *If Θ(f(x)) < Θ(g(x)), then the limit of f(x)/(g(x) as x approaches ∞ is 0. (This will be the case for (n!/nⁿ).)*
· If Θ(f(x)) > Θ(g(x)), then the limit of f(x)/(g(x) as x approaches ∞ of f(x)/(g(x) is ±∞, depending on the signs of each function's limit.
· If Θ(f(x)) = Θ(g(x)), then the limit of f(x)/(g(x) as x approaches ∞ of f(x)/(g(x) is the quotient of their 'asymptotic coefficients'.

Basically, what this means for the limits of quotients between polynomials as x approaches ∞ is that when applying one of the limit-of-function-quotient rules I listed, *everything minus the leading term of both the numerator and denominator can be omitted, which is incredibly efficient.*

Inspirator_AG

This is also known as Sandwich Theorem.

rishavchoudhuri

I have not even taking calculus yet (However I take Calc I next semester), and I find your videos really easy to understand, except for a few calculus concept

crosisbh

Your explanation is very simple and easy to understand for even me, Japanese 🇯🇵😆

dagajsb

Consider the following generalization:

We have lim n-->inf of (n!*k^n)/n^n, where k is a positive real constant. You looked at the case k=1.

In fact, for some values of k, the series diverges while for other values, it converges to 0. For what value of k does the series flip from converging to diverging?

ExplosiveBrohoof

I had that exact problem on my maths assignement this week :)

sUpErTrEkKiE

on the top, you end up with:
n^n -(sum of k up to n) *n^(n-1) +(sum of [sum of j] * k) *n^(n-2)...
already, sum of sums is roughly sum of quadratic, which is cubic, and this is really difficult to work out because the first term is the only thing of degree n, all after are n+1
good thing for the squeeze theorem

MrRyanroberson

"The list" is very useful for anyone studying Big O Notation.

AdamDavis

Your videos are very interesting and useful! And, you know, as you mentioned best friend, the fact and the list I started wondering. Are all your videos/lessons a preparation before some gigantic maths problem to solve witch we will need all this knowledge?

plchlog

I'm a student from Mexico, and I finally found help!

josemanuelalvarezguzman

I am pass all that, but I wonder how my teachers would have reacted if I wrote down "By Best Friend Theorem".

I did one solve an integral "By Table" and got a bad grade... I think my table as too good for my teacher's taste.

The book was McGraw Hill Schaum's Mathematical Handbook of Formulas and Tables by Murray R. Spiegel - not sure of the edition, it was one of those cyan cover.

Theraot

That squeeze theorem really fucks me up, great video thought

rodrigosuarezcastano

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