Limit as an integral (Riemann Sum)

blackpenredpen

[zeroregretsgiven]

From all I've watched of your vids, this one was the most amazing <3

[robincalmard]

Hi Black pen and Red pen,

I'm Robin from France, and I wanted to tell you that I love your videos ! Thanks to you I got better in analysis and had very good grades for my exams, and I wondered if you'd consider doing more probabilities / vecotrial, linear/ bilienar algebra problems ?
I wanted to share with you and everyone one probability exercice that I find pretty fun :

p balls ranging from 1 to N are drawn with replacement. Let X be the random variable that is the maximum (biggest number) of the p balls.

1)Describe (omega, P)
2) Find P(X = k), k ranging from 1 to N (hint : consider the event (X <= k) )
3) let Un = sum from 1 to n of k^a. Show that Un is equivalent to n^a+1/(a+1) when n approaches infinity
4) Find an equivalent of E(X) when N approaches infinity



Sorry for bad english, keep up the great work !
#Yay from a french fan,
Robin

[alkankondo]

#YAY! I was able to figure this one out quickly because of another video you did a while back in your "Math for Fun" playlist. Now, when I see a limit of the form "0+0+0+...+0, " I start by looking to see if I can turn the limit into an integral. So, with this limit, once I realized it was an integral, I just wrote down the definition of the integral as a limit, then found the Δx, the x_k, and the bounds of integration. It was easy from that point; I did the rest in my head!

I love your sense of humor (e.g. the "awkward" beginning of the video). I bet your math classes are loads of fun!

[laugernberg]

I like your explanation of what intuiton one should have. But i also want to submit a perhaps more formal answer:
- infinitely 0's added is indeed 0. But this is a series, and each term approaching zero doesnt imply convergence, and specifically not convergence to 0, (in this case it is convergent though) :)

[Sid-ixqr]

So today I was called up by my mathematics teacher in school to solve a math problem on the board.


I realised how hard it was to write on it. My writing turned out to be in shape of a curve.... Which you just integrated!!

#YAY!

[JoshuaHillerup]

Wait, we didn't do integration until Cal II.

[ghosttwo]

Thanks to you, I now _hate_ math. Took 25 years, but you did it!

[Invalid]

Another awesome video/explanation.
Black pen, red pen yaaaayy!!! ☺

[karstenmeinders]

Very smart to determine the identity of this limit sum to an integral of a square root so it becomes a no brainer in the end! Very admirable!

[vilius...]

damn, this channel never ceases to amaze me

[jasperh]

Did i hear more math videos in the summer? Now that's worth a #YAY!

[MA-bmjz]

Sum from k=1 to n of 1/n f(k/n) converges to integral from 0 to 1 of f(x)dx

[jlxip]

This is the first video of yours I was able to do before watching

[gabest]

Tried to apply this for 1/x from the last video: => 1+1/2+1/3+...1/n. It does look divergent.

[es_for]

Guys this is why he takes upper limit as 1. If you know riemann sum, your width of each rectangle is what? Well it's whatever length you have divided by how many rectangles right? Well 1/n is saying you have a width of 1 of the total function (so upper limit 1 lower limit 0) and divided by n rectangles

[theominousu]

actually we can use squeeze theorem to do this problem as well

[cutecommie]

Ok, but why is the integral of √x from 0 to 1 equal to 2/3? What is the antidetivative of √x?

[digaddog]

My teacher asked us to find the integral by taking an infinite reimann sum. Is that a thing?

[RomanNumural]

Do you think it would be reasonable to attempt a Delta epsilon proof here?

Also could you do a dummy variable substitution letting n=x where x is a real number and then factor out 1/n, Then use L'Hopital?

Could somebody please explain to me why does he take as an upper limit 1? Is it by definition or some property?