i think you typoed at the end, should be ->1,but very clever way of doing this, awesome, thanks for sharing, I don't think I've ever seen it done this particular way:)
3:03 may I ask why is it that we're able to insert another equation in between and use that one instead? edit: (my thinking), by an equation larger than the original one, wouldn't its N also be larger? and shouldn't we find the lowest possible N?
You explained this very well, and in a fun way! I always get stuck on proofs. Does this show it converges then? And do you have any videos that explain what it means? Thanks!
ya it shows it converges, yeah check my advanced calculus playlist for other convergence proofs, basically it means that when n gets really really big, n/(n + 1) gets really really close to 1:)
so if we let n go to infinity and we get a number, then we say the limit exists, it is the number, in this case 1, and the sequence n/(n + 1) converges to 1
For proving the sequence converges with the formal definition, how would we know when to stop, would be when we reach 1/n in general or varies by each question?
Is having the actual value of the limit necessary to be able to prove it's existence? I mean can you prove the existence of the limit _before_ finding it's numerical value?
Suppose you didn't use the fact that 1/(n+1) < 1/n then you would find n > 1/ε - 1. It will yield the same result at the end. My question is why did you use 1/n > 1/n+1 ? It seems unnecessary. And i am asking this because i have seen this proof in many textbooks done the same way you did it. Am i missing something ?
no you are not, it's just that , I think that n > 1/e is cleaner than n >1/e - 1. You could CERTAINLY do it that way, there is nothing wrong with that, you could say you are ch oosing an integer n such that n > 1/e - 1 via the archimedean property and it's all ok
But why is a convergent sum like 1/n regarded as =1, although it never becomes 1 ? Is it by Definition regarded as 1? Or is it a believe that it becomes 1 by Infinite procedure?
What would go wrong if I tried to prove that the limit converges to any number other than 1? What would go wrong if I tried to prove that it converges to 2?
At 3:10....How do you know that 1/n is less that epsilon? Since at 4:58 you said we can't just write |(n/n+1)-1| < ε ...please explain this to me....and what does everyone mean when they say they can "choose" epsilon?
because we are figuring it out(it's the scratchwork, so we can do anything), in the proof, the 2nd part, we have to formally show it:) Sometimes people say choose e>0, but really,it's N that we choose. In these proofs we have to find N. I will upload more of these soon. It's a tough topic!!! BTW the reason we can choose N is because of the Archimedean Principle, it says given any number c, we can find a natural number N that is bigger than c. So in this example we chose N > 1/e
my teacher almost always uses N interchangeably for integers. Like why why why... and he wrote his own textbook rather than using something excellent like Cummings or Buck.
Why do you have to show every step of your simplifying? Why can't you jump from the definition of convergence to (1/n)? (Trying to save some pencil lead) :D
Great explanation! One part I got confused on though, why exactly did you need to apply the Archimedean principle and choose an N greater than 1/ε? Couldn't you just keep N = 1/ε, since we're going to end up with the n being greater than 1/ε anyway? Is this because we want N to be a positive integer, and if ε ∉ Z+, we will just have 1/ε ∉ Z+?
