Proving a sequence converges using the formal definition | Series | AP Calculus BC | Khan Academy
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Applying the formal definition of the limit of a sequence to prove that a sequence converges. Created by Sal Khan.
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note for Real Analysis students: M has to be an integer, but by the Archimedean property of real numbers you know there exists an integer bigger than 1/epsilon. So you can set M to ceiling(1/epsilon).
Just putting it there for people who need slightly more rigor
thanks i was confused in the video by now you made it clear
so after watching many 2 hr videos here comes this 5 min video which cleared everything💐💐
Analysis fails to make sense to me. I literally only pass because I cram the formula solve it to get an answer in terms of epsilon, rewrite the proof backwards and end up with the formal definition.
Did I prove that the sequence converges? Apparently, I did. Do I understand what I just did? No! Not in the slightest bit. But hey, at least I got it right.
thank you for this comment
I'm taking Real Analysis right now, not understanding the thing but memorizing the steps from sheer effort.
@@lemyul SAME
Late response, but for maybe you and anyone else having the same question, here's the answer.
We just proved that for ANY epsilon, the nth term of a sequence is Epsilon places away from our limit.
This intuitively doesn't make sense. But the REAL power of epsilon being anything is that I can make Epsilon smaller and smaller. So M becomes bigger and bigger.
What's happening to the sequence though? Now the elements are going closer and closer to the Limit as epsilon becomes smaller. So the sequence "TENDS" to the limit.
I could make epsilon INFINITELY smaller with impunity. That's where the essence of " taking a limit" starts to peer through the seemingly abstract epsilon concept
@@saurav3078 I love you
literally still confused 😢😥
Niya Tin why?
5:16
well you graduated so congrats
rewatched this 4 times before the last part made sense. But eventually everything made sense! Thank you!
Dominic Cordeiro mast
OMG. I just saw this proof so clearly, for the first time, and I could almost consider hugging you-- I am that joyous. Thank you KA!
less than episolon < means that value will be within the episilon value (aka the green line of the y axis.) ...so basically all those dots that are ahead of your current x value will lie within the green part............easy to visualise as the dots oscillating back n forth and gradually lose energy and settle on 0
i don't get why my teachers take hours to explain nothing and this takes 5 minutes to explain everthing I need. thx!!
Thank you so much for this amazing video! When I first learned this knowledge in my first-year calculus class, your crystal-clear explanation helped me a lot! Now I am trying to refresh my memory to better prepare for my ODE class next semester. This video still helps a lot! Thank you so much!😀
Sweet! Going through Limits of Sequences and have not at all heard of Epsilon nor M. Even more confused! Awesome!!!
Wow. Great video. This explanation toke a heavy weight over my brain. Its so much helpful when you use graphical illustrations.
epsilon is a margin of error. It is literally just an arbitrary small number. What's important is that by this definition, no matter how small the epsilon you choose, for any points after M, a_n will be smaller than epsilon.
very help full for understanding converges sequence
I think its vital to state that the choide of n>m is the index of the series? So both m and n have to be natural numbers?
+Qlimakz only n has to be a natural number since it's the index of the series. but M can be any real number since it is a function of epsilon and might not always be a natural number. hope that helps.
but isnt the function, or the series discrete? We only get outputs for discrete inputs? So choosing an m of say 2.5 wouldnt make sense would it? im probably missing something...
The indices are discrete but the M doesn't have to be. The M is just a number where if the sequence converges, the index is larger than that M and the terms after that index in the sequence are within some distance episilon from the limit.
So if M=2.5 then everything from a3 on will be within epsilon of the limit.
That distance could be 0.00001 or 1 vigintillion depending on how the sequence is constructed.
Oh I see, because the definition says n>M, which implies it would be discrete anyway?
Very helpful in explaining this abstract topic. Have cleared my long-standing confusion. Bravo!
Thank you so much..this was the best video explaining the concept clearly!
You're the best khan academy!
I just love u Sal khan. this was amazing cant be explaind any easier then that
Thanks sir !!!
You did help me in Midterm crisis. Thank you so much!
Wow...
Thanks for this sir. It helps
If epsilon is less than zero then it will still be within the bounds of +/- epsilon, i.e. if epsilon is -.5, then you're looking at a band between -(-.5) and -.5
if n is even, there is a negative value in the modulus, hence -1/n < €, although that should always be true since € > 0. Can't understand why you treated the -1^(n+1) as 1 for every n.
+Alvin the Chipmunk Because he took the modulus of that function. Mod(-1)=1 so when (-1)^(n+1)=-1 The modulus flips it to 1. Therefore we treat (-1)^(n+1)=-1 as 1
absolute value
|-1/n| = |1/n| = 1/n < epsilon
Khan Academy is a life saviour,Thanks and keep more videos coming
Best explanation.
Great explanation sir...
M cant be equal to 1/E, because N has to be a natural number, but N can be equal or greater than 1/E
he states in his definition that it is any M>0 (part of the reals), and n>M (n is a natural number so any natural bigger than M) so hes okay. (n cant be equal to 1/E btw because then in this case; mod(an-0)=E
For the fast time maybe I didnt understand a Khan Academy video at the first try
oh. very interesting lecture!
please post more videos on convergence
The best 👨🏫 ever
thank you bro !
I feel khan has derived his style from princeton prof adrian banner .prof adrian banner videos on calculus and linear algebra are there and they are great
thanks.
I think I’m falling in love with you again khan academy
sir, but what if I want to show that "M" is a natural no.?
i love you!!!!
Would have been a little more clear with the correct notation, instead of m=1/E, it should be m:=1/E since we define it in this particular case. Nonetheless: Great video !
Why do we need M?
Please clarify my doubt
Good video
because you're proving the limit for all n>M in the sequence.
If you're given an error tolerance (epsilon) you're showing that past the Mth placed element in the sequence you can get within this error tolerance.
Please explain about m
Who else watches these at 1.5 speed?
I wish i could play it at at 1.75x
Great idea thanks!!
what is M the output?
Let epsilon be less than zero.
but a(1)=1 and it is not in the region mentioned... can someone please explain?
Ahmad Naji Based on the epsilon value, the series does not start to converge until M = 2. a(1) = 1 is a value outside of the range defined by the limit +/- epsilon! At least I think so.
Huh?
M? Epsilon?
What is M stands for?
what does epsilon mean?
TerenceEB "an arbitrarily small positive quantity is commonly denoted ε"
I don't understand what's going on here at all
Dambo Jamilla look at d definition of a sequence.
Coz u a "dumbo"
Why m = 1/e
How do you prove that the limit is wrong?
I don't get what's so hard to understand. Most people watching this are probably taking analysis and I can only assume you guys have done epsilon-delta limit proofs. This is pretty much the same thing. The goal is to start backwards and find an M. The proof only starts after you know what M is. You can prove what you want to since you know what M is
Solving the proof (to get n > 1/E) is not always this easy for every question.
Can we have a more complex example?
Its just a greek letter notation
he didn't explain what he means by epsilon and M. I had to rewatch like 5 times and google stuff to understand what he's talking about
from 3:21, doesn't that mean you also proved that 1/n converges? 1/n diverges though...
1/n doesn't give enough information to determine whether something diverges. If you want to analyze the sequence, for 1/n from for eg n=1 to n approaches infinity , yes it diverges. However, we could take a finite sequence instead, which converges. 1/n is simply just a number in this case, not a series or a sequence. So there is no question about divergence.
Ooooo
varenci, what if you choose epsilson = 0.01, clearly it is larger than most of the a.n terms??? help:(
Epsilon is an arbitrary number that is becomes in question. Many of the people here are missing the point. You do not get to choose epsilon. Epsilon is a distance that is given to the particular question you are asking and it is always greater than 0. The point you get to choose (in higher mathematics) is delta. This delta says that given any Epsilon greater than 0 you can find a delta greater than 0 (no matter how big or small that delta needs to be) that will work for that particular epsilon. In english it means no matter what value epsilon you are given you can ALWAYS find a distance in a convergent sequence which if a distance of two points is less than the delta you create than the distance of the sequences corresponding to those points in the sequence will be less than epsilon.
Thank you so much for your reply, i understand better what epsilon means in the definition! very much appreciated!
Mike Fritz Why can't you choose epsilon? It's like you want to know for what n and M the distance between the sequence and the limit has a particular value, say 0.01. In the video it is shown that n > 1/e, so by filling 0.01 in for epsilon, we get that the sequence will converge when n gets bigger than 100. I can't see why this would be wrong.
borisjo13 Epsilon is an arbitrary number such that epsilon can take on any positive number from 0 to infinity, and An < epsilon, for any epsilon more than 0, it means that An will tend to 0 since Epsilon can take on an infinitely small number like 0.0000...0001 and An will still be smaller than that.
Who comes up with these definitions? Maths is not about words, but pictures and formulas. I could finally see what they mean with all those Ms and ns and what not.
Oh sorry I meant to say n. I watched the video without audio lol. I'll delete my comment
this guy says the same thing three times like for everything he says its really annoying