how to use the ratio test for the series of n!/n^n

We will determine if the series of n!/n^n converges or not by using the ratio test.

subscribe to my new calculus tutorial channel @bprpcalculusbasics

[kingarthr]

man anything from calculus 2 gives me good memories of december 2019 and just how great things were before 2020.

[alkankondo]

The example problems you chose to work out are always so rich with things to learn from them. I always come away smarter after every video!

[tyrannism]

I like the questions that are like "Use the ratio test to-" and then immediately answering "convergent" and moving on to the next question.

[nathanielsharabi]

love these videos man. even though i'm 17 and only doing basic calculus in school with your videos i'm now able to follow along with the solutions you make and get a peak into the harder stuff i'll do later on 👍

[saravigario]

you're accent is cute and you're very incisive and to-the-point. Thank you for making content!

[kaitlynleffler]

omg. tysm. I've spent at least 10 minutes trying to solve this one problem and i didnt even see there was a definition of e hiding in there

[nicolaspaglione]

188k people had the same problem and he gave a quick and full answer imagine how much time he is saving us collectively

[ffggddss]

Neatly done!
Next question this begs is, what does it converge to? It's probably nothing intelligible.

[Krystaltho]

I helped my friends with this problem a while back and it was really fun. I did it a bit differently but got the same result

[colorfulcalculus]

I just had my Calculus 2 Final Exam, and a question about improper integral was asked.
Question:
Which of the following improper integral converges?
A. integral from 1 to inf of ln(x)/[ln(x)+x] dx
B. integral from 1 to inf of e^(x)/[e^(x)+e^(-x)] dx
C. integral from 1 to inf of (x)^(1/2)*e^(x) dx
D. integral from 1 to inf of e^(sin x)/x^2 dx
E. None of the above
I hope you can make video for this question. Thanks :)
p.s.: I answered D

[cassied]

5:15 as soon as he flipped the fraction I saw where he was going and my mind was blown... I would've never thought of that

[cassied]

I just cant believe the dexterity with this guy... holds two pens in one hand a mic in the other.

Others have probably commented this same thing on a number of his videos but I hadn't seen it with my own eyes yet, so I had to say it XD

Great content!!

[ai_serf]

that algebraic transformation to 1 + a/n was pure finesse.

[haydenskelton]

Your videos are so great!!!! This problem was tripping me up so bad on my calc 3 homework! Thank

[gregorio]

Another way to do it is to see that for n>1, each term is less than 2/n^2, so this sum is at most 1+2*(zeta(2)-1)=2, 29

[sevret]

I'm glad I looked up how to derive the formula for the Poisson distrubution, as that uses many of the similar tricks in this example which makes it simple to get.

[gillrowley]

You are insanely smart. I used to know this stuff. thank you for the math lessons

[BigDBrian]

You can tell intuitively that the terms get smaller and smaller (note, doesn't necessarily prove convergence, see the harmonic series!)
n^n takes n and multiplies it by n, n-1 times.
n! takes n and multiplies it by numbers smaller than n, n-1 times.
It's the same amount of multiplications but the terms get smaller.
Therefore n^n will be bigger for arbitrarily large values.

[drv]

Your each video is a masterpiece .Thanks for such content 😊

[AntoinetteDavido]

Thanks for your help! Made my homework question more understandable!