The p-integral Proof (type 1 improper integral)
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- เผยแพร่เมื่อ 23 ก.ย. 2024
- 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.
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I showed this to my cat. He looked at me funny and walked off.
Vecht
Oh mannn!!
cade
He doesnt like it when Steve uses the blue pen
Lol
Should have pawsed it for him
just wanna take the time to thank the great educational job you do here
Fractal thank you!!!!
The hood went up.. you know things are about to get real.. or complex... or non-elementary... you never really know with calculus.
maybe things are getting series
Hi~
Hi
wanna ask how to integrate indefinitely of 1 over cos (2x) minus cot (2x)
. im a beginner
Hi!
Hello!!!!!
Say.Is it Peyam?
I like doing pintegrals
So, you have a pinterest in them?
Fred
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
No joke, my cat just watched this video with me.
R.C. Whitehead nice!!!!!
Your cat loves BlackPenRedPen himself.
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?
You are correct. Look up the integral test for convergence.
pretty much
left a like and put in my Watch later playlist
Felipe Lorenzzon :)))))
Haha, that’s what I call the Peyam test in class 😄
Dr. Peyam's Show nice!
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
Is answer -875?
I get 125.
Pretty sure z=5exp(I*pi/3), sound right?
Sir you are like a lifesaver calculus angel. Thank u very much😊
You are such a good teacher!are you still running marathons?
Nacho I am. March 24th, LA marathon!
@@blackpenredpen YAY!!great to hear
@@blackpenredpen Why do you do that _AND_ the maths stuff instead of just one XOR the other? Is it NECESSARY to do both instead of just one XOR the other? Are you a BAD person if you only do one, but not the other?
@@mike4ty4 what?
This is lifesaving ! I know my cal II test tmr will have at least one question related to this !!!
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.
Know thy integrations and believe thy limits - Chain Lu (2019)
i appreciate this expression because you made it simple though it seems difficult!!!!!!!!!!!
This reminds me a bit of the geometric series or atleast some sort of special case of it. Is there a connection?
he explains this way better than my teacher did I finally somewhat have an understanding of this
This is a really good video man! Nice work!!
my go to youtube channel to pass this calc 2 course!
I still find it very counterintuitive that the p = 1 case diverges. The area under 1/x gets smaller as x goes to infinity so it looks like it should indeed converge. But I guess not...🤔
It diverges, but ve-e-e-ery slowly - logarithmically, in fact.
He's apparently going to show in a later video, how this integral ("p-integral") helps analyze the behavior of the p-series
∑₁⁰⁰(1/nᵖ)
using what is called "the integral test." When p=1, the series is called the harmonic series, and it too diverges logarithmically.
This is counterintuitive for the same reason you gave for the integral.
But in fact, there's a famous constant that is defined by having the integral and the series "run alongside each other" all the way to ∞.
The Euler-Mascheroni constant is defined by:
γ = lim [ ∑₁ᵏ(1/n) - ln(k)]
k→∞
Its value is about 0.577... (You may recognize this as the first 3 digits of tan 30º = 1/√3, but, alas, the two don't match after 3 digits.)
Fred
Hey, today I went to a math competition and there was this one frq that was really confusing. It had 2 equilateral triangles and a line going through. With those we are supposed to prove two lines intersecting are pi/3
At first it looked simple but then turned out to be seemingly impossible to prove.
I later asked about it and it had something to do with the complex plane somehow.
Would you be able to figure out what I am talking about and make a video on how to do it
You broke a lot of pure maths rules there mate!
Thank you so much, I was wondering what my professor was doing but now I get it.
I guess you could you exponent of convergence to find out for which power it converge which is equals lambda+1
Awesome video but I think you forgot to switch off AC !!........no?
Or Something alse?? 🤔🤒
....little help needed... How expand (4a+5b+6c+.....8e )===.......if a, b, c be the positive integers ?🤑
Thank you for such an amazing video! One question, does the first integral, where p=1, have anything to do with the p-value test (series)?
yes, he is demonstrating in the video how it is known that the p-series converges for p > 1
On the same way, serie from 1 to infinite of 1/n diverges and serie from 1 to infinite of 1/n² converges.
Just did this problem in class on Friday.
Habib Hassan nice!!
What about 1>p> 0 ? Is p an integer?
I know that it's not really 5he right video to ask under, but I really don't understand one thing. If you have a definite integral and want to solve it, by using a trig substitution, then you would also have to change the bounds of it, not just the inside. But let's say the integral is from x=-1 to x=1 and you want to bring it to the theta world. With the substitution e.g. x = cos (theta). So you have 1/-1 = cos (theta), then for both 1 and - 1 you have an infinite amount of theta's to choose from. Do you you always have to choose the principal values? How does the choice of one affect the other? Thanks to everyone in advance!
PS : I'm a 13 year old from Germany, so if I don't have perfect grammar or spelling or I have a flaw in my logic... Just don't be rude. :)
Hi, how can I send you a message? I didn't find any email addresses...
What I do not get is that how can that area diverge? It can obviously be seen that it gets smaller and smaller, how come it diverges? I understand all the mathematical operations except for this.
x--> infinity, lim 1/(x^(p)) -->0, if p>0 or p > 1, how about p=0.01 > 0, the limit will be 0, can you show me the proof?
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😂
blackpenredpen, great vid :D also, Im learning about sums and series in university and I need help. Are there any helpful videos on youtube?
thank you so much!
By the way, that integral is equal to 1/(p*ln(p)) for p
Álex the Benighted yup : ))))
p progression
1/x^p p=1 harmonic series
Thanks for another interesting video. While watching the video, it occurred to me I never understood why the integral of 1/x equals ln (x) ?
Mark Berardi
It's how they defined ln(x) first
Can you upload a video of the repeated exponential graph? It would be very interesting, greetings
More of these videos where you end up with a formula! :D
+blackpenredpen, Hello Sir! Greatly impressed by your content. I have a problem and will be highly obliged if you can address to it.
Let f: R--»R be a function which is continuous at 0 such that f(0)=1. Also assume that f satisfies the relation for all x:
f(X) - f(X/2) = 3{(X^2)}/4 + X. Find f(3).
Thank you.
Love from India!
Amazing explanation!
I am taking my first year of math for my bachelor of science. Do you think I should look out for complex numbers at all?
youll get it eventually, but if you want it you can
at least the intuition for them js important to have
@@98danielrayI've dealt with the likes on the square root of negative 1. However, trig like cos(z) = something. Yikes
Can you make a video that evaluate integral x/(e^x-1)dx x=0 to infinity?? Plz
U are a machine and i like so much your videos
Nico Rojas thanks!!!!
Hi blackpenredpen I am mohammad Daghimen can you make a vedio to how we can Integrate the cube root of(( (x^3_1) ((over x_1 ) ) I love you alot LOL your channel is my favorite on TH-cam
continue bro you are great
Good job bro from Morocco
Shouldnt there be a 3rd case where p=0?
Thankyou
Isn't this the zeta function?
thank you
thanks for making late night maths fun :)
Thanks a lot !!
Can p become negative number? Or is it true for only p>1?
Ayip Hikhok The integral diverges for p less than or equal to 1, so that also applies if p is negative.
@@XXgamemaster ahh thank you.
Excellent!
always believe in the math!! nice video broo
Very nice!
Is there very cold out there?? 😂😂😂
Watching this 5 hours before my real analysis exam (I'm not so cooked after all)💀😂
The man is sporting the hoodie on. 🤪
integrate of cos x/ x
by parts is cosx lnx but cosx/x^4 ?
The negative version of the best friend!
Hey guys i need help in an integral : integral from 0 to 1 to 1/(1-x) + 1/lnx dx . Thanks
diverges when p = 0?
X^(x-1) ?
Believe in your limits!
Once again, you've saved me :)
the drip is continous but not differentiable!
Type 1 or type 2?
P-Series
So if p is only infinitesimally greater than one, it converges to one over an infinitesimal, which is still infinity. Ok, got it.
Do p=i cmlx
Why is this an improper integral ?
The negative best friend aka the worst friend
PLEASE make complex version, formula for 1/(a+bi)^z
Ginger Stalin and the limits of integration?
the condition is the same
converges for Re(z)>1
@5 mins: or just add p to both sides you crazy man!
I screwed up my math interview in December because of this question.... damn
Matthew
What interview was it?
This is in my calculus book.
Dr. Chow is in ur calc book?! Amazing!
@@budtastic1224 who is chow?
@@luisramrod9121 bprp
hi. all. :-)
I showed it to my dog it run scared heeeeeeeee
matache broo i got it bro
Hi5
He's dope
my cat just not getting it....perhaps I need to show more times :)
I love you
你太棒了
Alpha Tauri thanks
U r very fast...
Mathematician are triggered seeing him plugging in infinity like a number instead of using limit.
❤❤❤❤❤❤
I like your videos, but please use ln(x) instead of ln|x| for the integral of 1/x. Please delete all of your videos and reupload them but without the absolute value.
physics physics why’s that? The second one is correct for a larger domain than the first
Jacob Jeffries I did use ( ) in this video. Since I know the interval is pos so no need for abs value.
@@blackpenredpen Still, that is only for this purpose. Please use ln(x) for integrals in general.
physics physics
How would you integrate 1/x from -3 to -2?
@@blackpenredpen -0.4055
Emmy good
Hi
Too easy.
Yeah
Yes
Next time do some exercises okay me bro
P-integrals on pinterest