I really like your tendency to draw meaningful pictures. A common pattern of behavior I see from high school students to math majors in university is not drawing pictures, and I honestly don't understand why, since not thinking visually just makes everything so much more difficult. I've recently been reading a measure theory textbook, and a lot of the steps are quite difficult to follow on their own (some definitions in particular seem completely arbitrary), but they just pop out if you draw a nice picture. I really think it's difficult to overstate just how valuable visual intuition is when it comes to understanding abstract things; some people seem to believe being rigorous entails not thinking visually, but I really don't get this dichotomy - it's just important to keep in mind that pictures are merely aids in our thinking and that we have to make real arguments instead of just pointing at the picture.
Professor, I recently stumbled upon a sequence i find pretty cool and haven't really seen or heard anyone talk about en.wikipedia.org/wiki/Aronson%27s_sequence Aronson's sequence is an integer sequence defined by the English sentence "T is the first, fourth, eleventh, sixteenth, ... letter in this sentence." That's so wild to me! Hope you enjoy! Thank you!
Dear Dr Peyam. The Mean Value Theorem states that if f(x) is a differentiable function, then between any two values x = a and x = b there is a value x = c such that f'(c) = average rate of change. That is, the tangent at f(c) is parallel to the line through points f(a) and f(b). It is worth noting that: i) For f(x) = x^2, c = (a + b)/2 , that is, the arithmetic mean; ii) For f(x) = 1/x, c = sqrt(a b) , the geometric mean; iii) But, for what function f(x) would c = (2 a b)/(a + b), the harmonic mean? I couldn't follow the book* by Sahoo and Ridel, which by the way relate f(x) = ln(x) to the identric mean, beyond other generalized means. So I humbly request your help for (iii). Thanks in advance. (*) "Mean Value Theorems and Functional Equations" by P. K. Sahoo, T. Riedel.
Dr Peyam, you have done recently a lot of works on limits. I personally am very confused about hyperreal numbers and how they got postulated and more importantly why they came object of study. I personally refuse the notion of hyperreal number, but of course if serious mathematicians accept and use them, there must be something. I would be very grateful if you could shed some light on the subject. Thank you.
just one question at 7:40 how do we ensure that s- € is in the set S of all elements of sequence? As for the definition of the sup to work the element smaller than should also belong to the set of whose sup we are considering.
Corollary: If a_n is monotone decreasing and b_n is monotone increasing and a_n ≥ b_n, then each sequence converges. Proof: a_1 ≥ a_n ≥ b_n ≥ b_1 and thus each sequence is bounded and convergent.
Thanks for this lecture. I have a question in geometry . What are the dimensions of an equal sides Pentagon (upside down ) inscribed inside a triangle whose sides dimensions are 77, 106, 113 . Knowing that the individual angles of Pentagon measures 108 degree and the summation of angles are 540 degrees. with my best regards .
This would also work is a(n-1)=) for all n. Also, I know you pre recorded these videos but it would be cool if you made a video proving that every sequence has a monotone subsequence. Theres a really cool proof involving peaks for that fact. Then you get Bolzano-Weierstrass theorem for free!
Professor, I was given the following sequence: 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 21, 25, 26, 29, 32, and I was told that the next 5 points were: 33, 34, 36, 37, 40. I was unable to make sense of this. Could someone make sense of it?
Dr. Peyam, you should make some fun videos once in a while as well. While Real Analysis is good, most of the time we're proving simple results which are intuitive anyway Love your attitude towards math 😄👍👍
Professor...I face slightly difficulty in definite integration. Can u help me by some videos ? (Full definite integration) it will be really helpful. Love from 🇮🇳
@@drpeyam aah... Many mathematicians, programmers, geeks, etc have the same personality as people with Aspergers' Syndrome. Just wanted if you know anything about it....
@@drpeyam hmm if you find a free time try this test about Asperger's Syndrome: psychology-tools.com/test/autism-spectrum-quotient Its result may be interesting for many mathematicians..
Dr it was an amazing proof video, but i have a question remaining if it is possible for you to answer it . ¿ How do I find the bounds ? and why should i find them to see if a sequence is convergent besides just taking the limit?.thanks for the content.
I mean the bounds depend on the problem, but usually you do |sn| and do the triangle inequality. And it’s important in practice to see if a sequence is convergent, especially for proofs
@@drpeyam thanks a lot. About the triangle inequality, i've got one last question if you dont mind. do you know of some examples of functions that could be bounded with the triangular inequality, i googled it and i found its aplications in geometry and linear algebra but calculus.
I really like your tendency to draw meaningful pictures. A common pattern of behavior I see from high school students to math majors in university is not drawing pictures, and I honestly don't understand why, since not thinking visually just makes everything so much more difficult. I've recently been reading a measure theory textbook, and a lot of the steps are quite difficult to follow on their own (some definitions in particular seem completely arbitrary), but they just pop out if you draw a nice picture. I really think it's difficult to overstate just how valuable visual intuition is when it comes to understanding abstract things; some people seem to believe being rigorous entails not thinking visually, but I really don't get this dichotomy - it's just important to keep in mind that pictures are merely aids in our thinking and that we have to make real arguments instead of just pointing at the picture.
Yes while visual proofs might not be very _rigorous_ , they are very helpful for developing intuition
Professor, I recently stumbled upon a sequence i find pretty cool and haven't really seen or heard anyone talk about
en.wikipedia.org/wiki/Aronson%27s_sequence
Aronson's sequence is an integer sequence defined by the English sentence "T is the first, fourth, eleventh, sixteenth, ... letter in this sentence."
That's so wild to me! Hope you enjoy!
Thank you!
That is really cool!!! Thanks for sharing! 😍
An elegant explanation of an elegant theorem!
Want To Find (WTF), well played Dr. Peyam...
Dear Dr Peyam. The Mean Value Theorem states that if f(x) is a differentiable function, then between any two values x = a and x = b there is a value x = c such that f'(c) = average rate of change. That is, the tangent at f(c) is parallel to the line through points f(a) and f(b).
It is worth noting that:
i) For f(x) = x^2, c = (a + b)/2 , that is, the arithmetic mean;
ii) For f(x) = 1/x, c = sqrt(a b) , the geometric mean;
iii) But, for what function f(x) would c = (2 a b)/(a + b), the harmonic mean?
I couldn't follow the book* by Sahoo and Ridel, which by the way relate f(x) = ln(x) to the identric mean, beyond other generalized means. So I humbly request your help for (iii). Thanks in advance.
(*) "Mean Value Theorems and Functional Equations" by P. K. Sahoo, T. Riedel.
Dr Peyam, you have done recently a lot of works on limits. I personally am very confused about hyperreal numbers and how they got postulated and more importantly why they came object of study. I personally refuse the notion of hyperreal number, but of course if serious mathematicians accept and use them, there must be something. I would be very grateful if you could shed some light on the subject. Thank you.
Perfect Explanation
just one question at 7:40 how do we ensure that s- € is in the set S of all elements of sequence? As for the definition of the sup to work the element smaller than should also belong to the set of whose sup we are considering.
It’s not necessarily, the elements smaller do not have to be in the set
@@drpeyam Thankyou so much for your reply sir. after watching your lecture series I have been enjoying solving the exercises.
Corollary: If a_n is monotone decreasing and b_n is monotone increasing and a_n ≥ b_n, then each sequence converges.
Proof: a_1 ≥ a_n ≥ b_n ≥ b_1 and thus each sequence is bounded and convergent.
Nice
@@drpeyam I learned this as I was studying a proof of Stirling’s Formula.
If Modern Greek is bounded, then it converges. Ancient Greek, we're not so sure. A circumflex sequence might wander between bounds without converging.
Thanks for this lecture. I have a question in geometry . What are the dimensions of an equal sides Pentagon (upside down ) inscribed inside a triangle whose sides dimensions are 77, 106, 113 . Knowing that the individual angles of Pentagon measures 108 degree and the summation of angles are 540 degrees. with my best regards .
Ok. Thank you very much.
This would also work is a(n-1)=) for all n. Also, I know you pre recorded these videos but it would be cool if you made a video proving that every sequence has a monotone subsequence. Theres a really cool proof involving peaks for that fact. Then you get Bolzano-Weierstrass theorem for free!
Check out the playlist
Hey what happened to Steve from black pen red pen?
?
@@drpeyam he and fermatica is Friends with you and they haven't uploaded videos in a while
@@adityachk2002 bprp has taken a break from youtube since he uploaded the last video
cuonghienthaoson buitrung did he say that?
Professor, I was given the following sequence: 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 21, 25, 26, 29, 32, and I was told that the next 5 points were: 33, 34, 36, 37, 40. I was unable to make sense of this. Could someone make sense of it?
I don’t see a pattern either, but isn’t there a website for number patterns?
Dr. Peyam, you should make some fun videos once in a while as well. While Real Analysis is good, most of the time we're proving simple results which are intuitive anyway
Love your attitude towards math 😄👍👍
Real analysis IS fun lol
@@drpeyam Yeah but not _as_ fun as integrals 😏😏😁
Analysis is much better
How is sn-s < epsilon ?
It is negative and epsilon is positive
I'm indian🇮🇳🇮🇳🇮🇳🇮🇳🇮🇳🇮🇳🇮🇳🇮🇳🇮🇳 so this theorem is very important in bsc sem.. 2
Professor...I face slightly difficulty in definite integration. Can u help me by some videos ? (Full definite integration) it will be really helpful. Love from 🇮🇳
Check out my integrals playlist
DEF: Monotonic - just one fizzy drink
Sir vedio uplode green function visualisation graphically reply
No thanks
@@drpeyam
Why
Because
Fundamental greens function ∆2
1 = sqrt(1)
sqrt(1) = sqrt(-1 x -1)
sqrt(-1 x -1) = sqrt(-1) x sqrt(-1)
= i x i = -1
1 = -1?
line 3 is the error but can someone plz why?
Sqrt AB is not sqrt a x sqrt b
Do you know about Aspergers' Syndrome?
What about it?
@@drpeyam aah... Many mathematicians, programmers, geeks, etc have the same personality as people with Aspergers' Syndrome. Just wanted if you know anything about it....
Oh ok, I don’t know much about it
@@drpeyam hmm if you find a free time try this test about Asperger's Syndrome:
psychology-tools.com/test/autism-spectrum-quotient
Its result may be interesting for many mathematicians..
Dr it was an amazing proof video, but i have a question remaining if it is possible for you to answer it . ¿ How do I find the bounds ? and why should i find them to see if a sequence is convergent besides just taking the limit?.thanks for the content.
I mean the bounds depend on the problem, but usually you do |sn| and do the triangle inequality. And it’s important in practice to see if a sequence is convergent, especially for proofs
@@drpeyam thanks a lot. About the triangle inequality, i've got one last question if you dont mind. do you know of some examples of functions that could be bounded with the triangular inequality, i googled it and i found its aplications in geometry and linear algebra but calculus.
Have you already made a proof of the least upper bound property? Asking for a friend
Yes, check out my real numbers playlist
Pandemic proof channel :)
M S T stands for Maladie Sexuellement Transmissible in french xD
Plausible.