make it clearer if N-underscore is explicitly written as depending on the chosen epsilon, ie, N-underscore(chosen epsilon). or say 'for each epsilon' rather than 'for all epsilon'
Hi, what I believe he means is that it can very hard to recognize whether a sequence converges or not just by looking at the definition of convergence. For instance, let's consider the sequence {s_n} with s_n=1+1/4+...+1/n².
@ not sure that’s a good example. Here Sn=1/n^2 So |Sn-0|=1/n^2 so for all n bigger than squat root of 1 over epsilon this will be less than epsilon and Sn converges to 0
@@Happy_Abe be careful that my example is different: s_n is equal to the sum of all terms 1, 1/4, up to 1/n² (s_n=Σ_{k=1}^n 1/k² if you are familiar with this notation). In this case it is neither obvious if the sequence converges, nor what is its limit.
@@nicoz5787 oh I misread your example. Yeah that’s the famous Basel problem to pi^2/6. But still, it’s not obvious to me that it’s impossible to find an N in terms of epsilon that bounds the difference less than epsilon. I know techniques showing convergence are more involved but to say it’s impossible is a strong claim
@@Happy_Abe that's exactly the Basel problem. In the context of the video, I think "impossible" should be intended as "impracticable". I mean, by the very definition of convergence, if a sequence converges then the N-ε argument must hold true. However, if we tried to apply it to prove convergence of the Basel problem, we would first of all need some intuition that the sequence converges to s=π²/6 and then find some suitable inequality to actually prove it, which I honestly deem rather unlikely (at least if one is not called Euler 🙂).
I don't really understand why we should care about there always being a subsequence that converges to the limsup of a sequence. Won't it be trivial most of the time?
Hello! "12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is" Can you please give the answer of the question or like make a solution? Plz🥺
Literally had a lecture about this today. Thank you Dr. Peyam :)
Yay!!!
One man's ceiling is another man's floor!
Voulez vous Cauchy avec moi....hahaha you're just really awesome Dr. Peyam
erm why does this say 4 years ago
@@acuriousmind6217 Perhaps, it's due to the video being unlisted, then republished at a later date.
Good to see you back 🤍
make it clearer if N-underscore is explicitly written as depending on the chosen epsilon, ie, N-underscore(chosen epsilon). or say 'for each epsilon' rather than 'for all epsilon'
@3:08 what would be an example where it’s impossible?
Hard I get, but shouldn’t it always be possible?
Hi, what I believe he means is that it can very hard to recognize whether a sequence converges or not just by looking at the definition of convergence.
For instance, let's consider the sequence {s_n} with s_n=1+1/4+...+1/n².
@ not sure that’s a good example. Here Sn=1/n^2
So |Sn-0|=1/n^2 so for all n bigger than squat root of 1 over epsilon this will be less than epsilon and Sn converges to 0
@@Happy_Abe be careful that my example is different: s_n is equal to the sum of all terms 1, 1/4, up to 1/n² (s_n=Σ_{k=1}^n 1/k² if you are familiar with this notation).
In this case it is neither obvious if the sequence converges, nor what is its limit.
@@nicoz5787 oh I misread your example. Yeah that’s the famous Basel problem to pi^2/6. But still, it’s not obvious to me that it’s impossible to find an N in terms of epsilon that bounds the difference less than epsilon. I know techniques showing convergence are more involved but to say it’s impossible is a strong claim
@@Happy_Abe that's exactly the Basel problem.
In the context of the video, I think "impossible" should be intended as "impracticable". I mean, by the very definition of convergence, if a sequence converges then the N-ε argument must hold true. However, if we tried to apply it to prove convergence of the Basel problem, we would first of all need some intuition that the sequence converges to s=π²/6 and then find some suitable inequality to actually prove it, which I honestly deem rather unlikely (at least if one is not called Euler 🙂).
Can you Say A book where has many interesting sequence problem
I don't really understand why we should care about there always being a subsequence that converges to the limsup of a sequence. Won't it be trivial most of the time?
Good sup
Mathvis made of everything, fun things and also things like lim-sup 😕
Hello!
"12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is"
Can you please give the answer of the question or like make a solution? Plz🥺