The p-integral Proof (type 1 improper integral)

This is one of the must-knows in your calculus 2 class, the improper integral of 1/x^p from 1 to inf. We need to find for what values of p will this integral converge. This will be closely related to infinite series later on.

[VechtMalthos]

I showed this to my cat. He looked at me funny and walked off.

[danielray]

just wanna take the time to thank the great educational job you do here

[xevira]

The hood went up.. you know things are about to get real.. or complex... or non-elementary... you never really know with calculus.

[abrarshaikh]

You can also prove that using P-series (zeta function)
As series is convergence for p>1 only, hence integral will be also convergence for same

[roderickwhitehead]

No joke, my cat just watched this video with me.

[srpenguinbr]

left a like and put in my Watch later playlist

[nolann]

he explains this way better than my teacher did I finally somewhat have an understanding of this

[wouterdeniz]

Know thy integrations and believe thy limits - Chain Lu (2019)

[beril]

Sir you are like a lifesaver calculus angel. Thank u very much😊

[thefallinglink]

I expected it to converge for p>1 because this problem looks so similar to the p-series convergence test, but as an integral rather than a series. I'm guessing the connection is how integrals can be defined by Riemann sums with infinite rectangles, which is an infinite series?

[likris]

This is lifesaving ! I know my cal II test tmr will have at least one question related to this !!!

[drpeyam]

Haha, that’s what I call the Peyam test in class 😄

[viridianatirado]

my go to youtube channel to pass this calc 2 course!

[hauler-hxgw]

Here's a problem given to me in a math competition I had today that literally no one figured out in time

Let lzl=5 and Im {z}>0. If the distance between (1+2i)z^2017 and z^2019 is maximized, then determine Re{z^4} + Im{z^4}


Reply for correct answer if needed

[rockyk]

This is a really good video man! Nice work!!

[PaGow]

I saw this video last year, before I was doing calculus and I would never guess that I would need it today😅
Man...life is funny😂

[TheNachoesuncapo]

You are such a good teacher!are you still running marathons?

[MyNordlys]

On the same way, serie from 1 to infinite of 1/n diverges and serie from 1 to infinite of 1/n² converges.

[NavyBlueMan]

Here's a question, I just thought of it and don't even know if it's remotely possible to solve, I haven't had time to address it so thought I'd just drop it here:

The sum from n=0 to infinity of the INTEGRAL of 0 to n of 1/n!

It reuses n 3 times and so I'm not even sure how someone could approach it but still thought it would be worth asking just for fun. Maybe change it from n going to infinity to 1000 or something a bit easier to approach.

[pachulee]

Thank you so much, I was wondering what my professor was doing but now I get it.