Real Analysis 17 | Cauchy Criterion

[@qiaohuizhou6960]

0:35 properties of series 2:22 Cauchy criterion 3:45 use completeness axiom to prove 7:25 necessary condition for the convergent series

[@Kevin-xs1ft]

thanks, i see you in every video

[@qiaohuizhou6960]

@@Kevin-xs1ft I think I finished the whole series ... I love this video series very much the only problem is I was sort of doing it on my own so I don't have homework or exam grading to test how well I learnt from it. :(

[@douglasstrother6584]

So far, Real Analysis seems pretty straightforward. Understanding the nuances of the definitions takes the most work. I'm interested to see what unfolds.

[@brightsideofmaths]

Nice :)

[@zachsiegel3650]

"it's not a restriction at all because the only case we really miss is when we have the same index here: (5:01). Why do we not need to consider the case where m < n?

[@brightsideofmaths]

It's the same thing. You can consider this case but it's looks completely the same.

[@dinglerdangler]

THANK YOU, YOU BEAUTIFUL PERSON

[@brightsideofmaths]

Thanks :)

[@thealphanigga7129]

How you say "calculate" fascinates me and keeps me interested.

[@brightsideofmaths]

Surprisingly, I just learned recently that I pronounce it wrongly. Only since then, I do it correctly as you can see in my recent videos :D (By the way: I find it quite embarrassing that, in all my teaching and research talks, I always used a wrong pronunciation for such a key word)

[@ledescendantdeuler6927]

@@brightsideofmaths hahaha

[@hipolitdobrohna8279]

I have one problem with a reformulation of being Cauchy sequence: for n>=m>=N and taking m=N, we have |s_n - s_(N-1)|

[@carl3260]

Agreed, I think the N (call it N_1) in the second line needs to be set equal to 1 + N_0 (where N_0 is the N from the first line). Letting n be the larger of \tilde{n}, \tilde{m} it then works out. So there does "exist an N ...", it's just not the same N as previously.

[@XexJJ051301]

I was also unsure about this. On the one hand I agree - it just simply isn’t intuitive to me why S_(m-1) works. However, I think making N_1 = N_0 + 1 could also have its problems - I’m unsure if the double implication would still hold; perhaps there would be a way to reduce N to 1 if we keep going backwards. Edit: upon further consideration, it seems there is no difference between imposing the restriction n>m as opposed to n>=m (since these terms are arbitrary). The notational difference is S_m versus S_m-1. In addition to this, if we make n,m>N and n>=m>N, then everything becomes a lot simpler to understand, and S_m-1 holds!

[@sayan2009]

Love from India. You are doing a great job. Please cover complex analysis, PDE, differential geometry etc. Also, if you could give sketch of the proofs of the major theorems (like open mapping), that would be great. Your lectures are crystal clear. Thanks a lot for uploading.

[@MrOvipare]

In the beginning you said something like "we can do this because the series are convergent" after explaining their properties (a) and (b)... This reminded me of the famous result that 1 + 2 + 3 + 4 + ... = - 1/12 which I always found suspicious. I know we can have this result by using the Riemann zeta function, which is an analytic continuation... I can accept that, but 0n the wiki of "1 + 2 + 3 + 4 + ..." they show that Ramanujuan expressed the sum of natural numbers as c, then multiplied it by 4 and expressed c - 4c, then solved for c to obtain c = - 1/12. Don't you agree that those operations were not really "legal" since he manipulates divergent series as if they were numbers? If we stayed domain of natural numbers, it seems to me that the result would simply not be true, or at least it would be "undefined".

[@brightsideofmaths]

I don't want to go into the details there but, of course, you are right. We cannot just calculate with divergent series like with numbers. However, this does not mean that there are not other extensions of definitions, operations, and so on that could be used in a meaningful way for divergent series. In this case, also the interpretation of the results is important. I will explain this in another video some time. Here, for real analysis, it is just important to note that convergent series fulfil a lot of calculation rules.

[@MrOvipare]

@@brightsideofmaths Thanks! Can't wait to see what's next in this series but also in your series on complex analysis :)

[@angelmendez-rivera351]

@@MrOvipare Well, an issue that is important to discuss in this topic is the semantics of the symbols we are using. What is the symbol string "f(0) + f(1) + f(2) + •••" supposed to refer to? Mathematicians agree that there is no clear, unambiguous, natural, intuitive meaning that the symbol string can be given. This is why such a notation is never used by modern mathematicians in their works. The string of symbols is unmathematical and has no meaning. This is the real problem with manipulating these symbols. When mathematicians discuss what we call "series", they are referring to a very specific object, and they are very careful with the notation they use. The education system, however, is at fault for being sloppy with said notation and engaging in inconsistent abuse of language, resulting in the fallacious association of these symbol strings that are unmathematical with actually well-defined mathematical objects that already have a designated notation for them.

[@angelmendez-rivera351]

@@MrOvipare If you consider a function f : N\{0} -> R, and you define s[f] : N -> R by (s[f])(0) = 0, (s[f]])(n + 1) = (s[f])(n) + f(n + 1) for every n, then s[f] is called the sequence of partial sums of f, which is a concept that was already discussed in this channel. It is also called the series of f, though people often abuse language and use the word "series" to refer to lim s[f] (n -> ♾) instead, which makes for some really awkward, confusing, and nonsensical phrases. If you consider f(n) = n, then naturally, we can agree that lim s[f] (n -> ♾) does not exist, and any calculation that implicitly works with lim s[f] (n -> ♾) as if it exists is just as invalid as a calculation that uses division by 0, because the reciprocal of 0 does not exist. However, as it is unclear what the symbols being manipulated in the Wiki actually mean, because they are symbols that are considered obsolete, it is not possible to actually argue that the calculations are valid or invalid. It is also important realize that while lim s[f] (n -> ♾) does not exist, lim T(s[f]) (n -> ♾) does exist for some transformations T, and these transformations are the kind of transformations that are useful to consider as well as natural to consider in many settings, and they also have the property that if lim s[f] (n -> ♾) does exist, then lim T(s[f]) (n -> ♾) = lim s[f] (n -> ♾). So while lim s[f] (n -> ♾) does not exist if f = Id, there is a class of such transformations T such that lim T(s[f]) (n -> ♾) = -1/12. There are also other ways of getting to -1/12 without relying on the limit of a sequence. This is the sense in which Ramanujan summation is said to "generalize series", and the sense in which it is appropriate to talk about -1/12 in the context of f = Id. Calling it a "sum", though, is misleading, although I also contend that calling lim s[f] (n -> ♾) a "sum" is misleading as well, for the reasons I already mentioned.

[@MrOvipare]

@@angelmendez-rivera351 thanks! I know Ramanujuan was a clever guy, so I doubt he was really subtracting infinities. I will check out his method!

[@belfiore1495]

I didnt understand ... why did zou change the series into that at 5:42 ?

[@brightsideofmaths]

This is the sequence we get when we subtract both series s_n and s_{m-1}.

[@bishwajitsarma5768]

If N=1 (suppose), then S_m-1 foe m=1 is not defined. Am i correct?