It's a great privilege to have the opportunity to observe with our own eyes concepts we consider abstracts. Thank you very much for these videos!

@EugeneKhutoryansky

Жыл бұрын

Thanks. I am glad you like my videos.

I remember hearing this song when I was a peasant on some land next to a castle

@willytheriot8439

Жыл бұрын

😂

Awesome, as always. It's like to dream after remembering carefully selected topics. Great job.

@EugeneKhutoryansky

Жыл бұрын

Thanks.

Hi Eugene, I have read your animal rights website! You are an amazing person - superior, incredibly kind, and intelligent, The true intelligence lies in helping us grasp the meaning of a concept at its deepest level, rather than merely teaching equations. This way, we can utilize that logical thinking to derive equations. I wish you all the best of luck! Your enlightening videos always blow my mind, and without your education, my understanding of general and special relativity wouldn't have reached this far although I still have a lot to learn. Thank you for everything!

@EugeneKhutoryansky

10 ай бұрын

Thanks for the compliments and I am glad you liked my website.

Thank you Eugene for this great explanation ❤ You deserve a golden medal 🎖

@EugeneKhutoryansky

Жыл бұрын

Thanks

It is always a good day when you upload a video. Thank you for making these!

@EugeneKhutoryansky

Жыл бұрын

Thanks.

Another neat trick is to simply show these harmonic series as Fourier sums with changing "wavelength" that depends on an arbitrary range (essentially the number of positive or negative terms you wish to sum in a certain order), and see where this function tends to at infinity. This way you are free to adjust the periods and hence choose any value as the convergent value.

thanks again! I’ve found some topics that might be animated into videos, such as craystal oscillator Dilutionrefrigerator norton’s theorem band gap reference anti-bonding orbital Reluctance motor Thedeflecting force really appreciate your work! by the way, which university do you teach in? i heard you’re a physics professor. i want to go to your university😂

@EugeneKhutoryansky

Жыл бұрын

I will add those topics to my list of topics for future videos. I already have a video where I talk about the band gap. This is my video on semiconductors at kzread.info/dash/bejne/mqaksq18dK-4fpM.html In reply to your question, I don't teach at a University. Somebody posted that on the internet, but they were just guessing.

@ionasiiwu815

Жыл бұрын

@@EugeneKhutoryansky sorry for that! but i don’t think bandgap reference is that simple, otherwise there’s no difference from a zinner diode, others say it is composed of ntc and ptc, but i can’t figure that out😂

Thanks for this video. I'm learning for finals right now and one of the topics I'm about to learn will be exactly this.

@EugeneKhutoryansky

Жыл бұрын

I am glad that I finished this video just in time for you.

Also make a video on the various infinite series of Ramanujan. Would be very interesting

The narrator's voice is mesmerizing and soothing.

@EugeneKhutoryansky

Жыл бұрын

Thanks.

Eugene is a straight double OG to the triple OG to the infinityle OG my G

Excellent! Superb videos! Thanks for sharing a wealth of knowledge.

@EugeneKhutoryansky

Жыл бұрын

Thanks!

6:06 infinity has never been so terrifying!

Spectacular video!

@EugeneKhutoryansky

Жыл бұрын

Thanks. I am glad you liked my video.

I'm finding myself excited every time for a new video on this channel. It's the latest Marvel episode.

@EugeneKhutoryansky

Жыл бұрын

Thanks. I am glad you like my videos.

I expect this channel to reach 1 billion subscribers now. What a presentation , pls keep adding videos .

@EugeneKhutoryansky

Жыл бұрын

Thanks!!!!! More videos are on their way.

Absolutely Convergent to Perfection Video. Magnificent

@EugeneKhutoryansky

Жыл бұрын

Thanks. I am glad you liked my video.

Another topic masterfully explained. Thank you!

@EugeneKhutoryansky

Жыл бұрын

Thanks for the compliment about my explanation.

Awesome video as usual. Great work!

@EugeneKhutoryansky

Жыл бұрын

Thanks for the compliment. I am glad you liked my video.

I have been following this channel since highschool, thank you so much for your work ❤️

@EugeneKhutoryansky

Жыл бұрын

Thanks. I am glad you like my videos.

I have been a Sub. since 2013. Keep up the good work!

@EugeneKhutoryansky

Жыл бұрын

Thanks. And thanks for being a subscriber for the past 10 years.

In the first 30 seconds I got a better understanding of why rearranging can change the sum than I ever got in college.

Great video, Eugene! Glad to know that you're still active.

@EugeneKhutoryansky

Жыл бұрын

I am glad you liked my video.

When the orchestral music dropped. I felt like the detective figuring out who Keysar Soze was at the end of The Usual Suspects.

@tim40gabby25

Жыл бұрын

Spot on :)

Пропустил один из выводов теоремы. Можно переставить слагаемые ряда так, что сумма будет конечной, но не будет сходиться к какому-либо числу.

literally learnt this for the first time last week in my real analysis course

*I love this channel!*

@EugeneKhutoryansky

Жыл бұрын

I am glad that you like my videos. Thanks.

This is so beautiful

@EugeneKhutoryansky

10 ай бұрын

Thanks.

Best channel on KZread bar none ❤

@EugeneKhutoryansky

Жыл бұрын

Thanks for the compliment.

How to do the inverse problem? How, given a number, we can make the conditionally convergent series approach that number?

it's also possible to rearrange the numbers so that the series does not approach any number or goes to both positive and negative infinity

@EugeneKhutoryansky

Жыл бұрын

Yes, I am aware of that.

@itsohaya4096

Жыл бұрын

💀

Spooky addition at a distance!

In the video, the spoken words during the interval from 4:28 to 4:39 and the interval from 4:11 to 4:21 are identical - I don't know if you meant to repeat those words exactly... It makes perfect sense of course, but it does sound a bit odd.

6:10 start everything is clear. Awsome. Keep Do It. Thanks.

Suppose you had a sequence with all even numbered terms adding to +inf (1, 1/2, 1/3, …) and all odd numbered terms adding to -1 (-1/2, -1/4, -1/8, …). The whole sequence would look like 1, -1/2, 1/2, -1/4, 1/3, -1/8, … Would this sequence still he conditionally convergent? I ask this because it’s stated in the video that for a conditionally comvergent sequence, both the positive and negative “components” sum to +inf and -inf, respectively. I appreciate any clarification. Thanks in advance.

@viliml2763

Жыл бұрын

such a sequence is divergent and no matter how you rearrange it it will always go to +inf because +inf-1=+inf

2:58 WOW! I see it, it's so beautiful, it's just like a log graph!

This shows that addition and subtraction operators when used in infinite sequences are not always order independent. Similar to matrix multiplication

Thank you. Great explanation 👌 Does conditional convergence happen with series that have every other term in negative sign

@EugeneKhutoryansky

Жыл бұрын

I am glad you liked my explanation. In reply to your question: No, a series might diverge, even every other term has a negative sign.

Crazy that similar to real life, numbers get tricky on a very small scale. Its a terrifying reminder that we are all just information

Thank you for your work

@EugeneKhutoryansky

Жыл бұрын

Thanks.

Amazing video!

@EugeneKhutoryansky

Жыл бұрын

Thanks.

Extraordinary background music

When a steam boiler is producing 1.389 pounds of steam per second, how much steam is in the water of that boiler in any one instant of time?

So how by multiplying by negative one does the series alternate between positive and negative?

Quality Content thanks for your video ❣

@EugeneKhutoryansky

Жыл бұрын

Thanks. I am glad you like my video.

Сильный канал. Лойс.

5:47 is this not also the same phenomenon for the previous absolute convergent series? (at 1:31)

so infinity - infinity = any element in the set that defines those equally sized infinities?

A fascinating math concept! Thanks for the new video. To both you and Kira

@EugeneKhutoryansky

Жыл бұрын

Thanks! And thanks for your support!

Came for the music. Stayed for the math.

hmm it seems to me that but by rearranging the numbers one breaks the property with which this sequence was generated in the first place what is represented now has nothing to do with the original symbol and what it referred to

For the first time ever, I'm early! Let's watch this video!

❤️🙏

+1 for the music

Nifty !

Mind boggling

thanks for video learning ..... i need some time pondering

que buena animacion y explicacion ,estaria bueno que estubiese tambien en español

What music tracks are used in the video?

@EugeneKhutoryansky

Жыл бұрын

All the music in this video is from the free KZread audio library, and the names of the songs are the following. Renaissance_Castle William_Tell_Overture_by_Rossini

at 3:29 is when i was lost, what does "converge absolutely" vs just "converge" mean?

@EugeneKhutoryansky

Жыл бұрын

"Converge Absolutely" means that it would converge even when we add the absolute values of each of the terms. That is, add them together when all the terms are positive.

Nice video, nice music, nice science

@EugeneKhutoryansky

Жыл бұрын

Thanks.

Great

@EugeneKhutoryansky

Жыл бұрын

Thanks.

very useful for calc2

if a sequence is in geometric progression and the common ratio between every consecutive number is less than 1 , then we can estimate the closet value of sum to infinity, right?

@TheBasikShow

Жыл бұрын

No estimation needed! If the ratio is r, the sum is exactly 1/(1-r) times the initial term.

@jesusnthedaisychain

Жыл бұрын

S = a + ar + ar² + ar³ + ... S = a + r * (a + ar + ar² + ar³ + ...) S = a + r * S S - r * S = a S * (1 - r) = a S = a / (1 - r) S = a + ar + ar² + ... a / (1 - r) = a + ar + ar² + ...

Amazing! The video took a theorem that was very weird for me and made it super intuitive. When you showed that conditional convergent sums were made up of 2 sums, it all clicked. Of course you can rearrange it! Of course absolute convergent sums won’t be able to have that property

@EugeneKhutoryansky

Жыл бұрын

I am glad my video made the theorem intuitive for you. Thanks.

That's fascinating. Because we have infinitely many numbers to add or subtract to, we always find sub-sums which help us reaching our goal number. Like the room in the Hilbert hotel. You always find another one by shifting the rest.

@abmagnitude7596

Жыл бұрын

Having infinitely many numbers is necessary but not sufficient. In an alternating series with absolute convergence you also have infinitely many positive and negative terms but you cannot make them sum up to an arbitrary real number (which is basically the definition of unconditional convergence and in finitely dimensional vector spaces it is equivalent to absolute convergence).

@SellusionStar

Жыл бұрын

@@abmagnitude7596 Thanks, your right!

Positive or negative infinity

I think what this proves more than anything is that "infinity doesn't exist". It's only a construct in our minds. Basically, the amounts that would balance things out are basically put off forever and never counted. It's like flipping a coin an infinite number of times, but you just put off counting all of the times it's "tails", or only count as many of them as you need. I actually started writing this before it was mentioned that some think that infinity is only a mathematical construct, but I do agree with those who believe that is the case. If infinity exists, it exists simply as "nothingness", and thus still doesn't exist, but no physical thing or series of objects can itself be infinite. And maybe dimensions like time and space might (or might not) go on to an infinite extent, but no matter how far you go, you'll never reach the end, they may be unfolding with no end in sight, but infinity has no end, and thus can never be counted "after reaching the end".

Reality is a trippy place, but I don't think it's trippy enough to contain actual infinite sets of concrete things.

A very very very tremendous appreciation-deserving video from you my mentor.... You taught a very difficult concept of infinite sequences and series in only 7 minutes... You are so talented... During the video I felt a lot of intelligence, intuition and love as people feel in their NDEs 😅 It was like If there is a God, he/she himself is teaching me mathematics... One thing that convinces me that NDEs are fake and only the illusion of brain is that no one ever have seen how God created the universe or how the big bang happened in any of their NDEs nor did anyone see the dinasaur era not even the vision of how the universe will end... But my mentor these is still something mysterious that we humans can not yet understand because I myself had some precognitive dreams but they were not exactly how I saw them but they were about 95% true... I wonder how brain saw peep into the future during sleep. Maybe its not due to some God but due to a phenomenon like Delayed Choice Quantum Eraser where signals can travel back in time.. Anyway my mentor! Thank you so much for this brilliant video... All my anxiety went away by watching your video... I am anxiously waiting for your video on why all spinning astronomical objects release jets of radiation and materials only form their poles and not from their equator or any arbitrary surface position.. When is your next video coming? I have a strong desire to meet you once in my life but I am not in USA... I really regret it... I am struck in a 3rd World Country. 😥 Anyway how long it took for you to learn Poser 3D?

@EugeneKhutoryansky

Жыл бұрын

I think that at least some NDEs are more than hallucinations, because afterwards they know information they otherwise could not have possibly known, such as the details of what their family members several corridors away were doing at the time. I don’t know when the next video will be ready, as I just finished this video today. In reply to how long it took me to learn Poser, that depends on how complex an animation I would need to do. I learned how to do very simple animations in just a few days. Thanks for the compliments.

@physicslover1950

Жыл бұрын

@@EugeneKhutoryansky you are Welcome my mentor. Recently, my uncle had an NDE when he suffered a brain hamirage (sorry for wrong spellings) in the hospital, and what I know is that during that brain hammarage the brain is not able to function properly but he remembered it as clear as a glass. Moreover, after recovery he honestly told us about that experience that It was a hellish one... But he didn't told the doctors about it... What I am afraid of is that he is a very good & kind hearted person so listening to his hellish NDE has confused me... Does the kind of NDE we will see not depend upon how good or bad we are? I mean good people can have bad NDEs and Bad people can have good ones? If this is true then this is very unfair and heartbreaking 💔... What's more heartbreaking is that my uncle didn't tell us about meeting their deceased relatives in his NDE... What confuses me the most are the japanese NDEs where they neither see any heaven nor God rather they see a river and after crossing the river they find themselves in a beautiful park... No one has ever seen or talked to God other than in an NDE... Does God equally loves all of us, even the bad ones & the tyrants/ bullies... If this is so, then this is unfair... Because Bullies are the worst kind of humans on this planet.

@EugeneKhutoryansky

Жыл бұрын

I don't know the details of what happens after we die. I just believe that at least some NDEs are real, and that there is therefore "something" after we die. What this "something" is, I don't know.

@physicslover1950

Жыл бұрын

@@EugeneKhutoryansky The most frightening thing about that 'something' is that It can be 'anything' ... I think the more we spread love in this life the more we will feel loved in the afterlife and the more we spread hatred, anger and tyranny, the more we will have a hellish NDE... What I believe is that God only created the Universe but he didn't created us.. It is the universe which created us... That's why there is so much suffering in this life (our universe) but there is absolutely no suffering in the afterlife (God's spiritual Realm). But I also use to sometime believe that the afterlife have nothing to do with God, if this life is possible without God, the afterlife can also be possible without God... Maybe the afterlife have nothing to do with God. (as suggested by japanese NDEs).. The thing which I am most afraid of is the concept of reincarnation... Like Carl Sagan said that the universe wants to know itself through humans.. I don't want to respawn in this world where there is a lot of suffering but if the universe wants to do this then I am helpless... There is only one hope that if I reincarnate I will see the Glorious future of the humans I will know more about universe and Quantum mechanics in the future world... I will know about the pictures captured and the information gained through james webb telescope and luvoir telescope and much more powerful observatories of that time... I would be glad to know that much about the universe.... What are your views on this ?

@EugeneKhutoryansky

Жыл бұрын

Some people claim to have met God in their NDE. Aside from NDEs, I have heard stories of children remembering past lives, and knowing details they could not have possibly known.

This video both blew my mind, and re-assured me in my finitist opinion. Well done!

@EugeneKhutoryansky

Жыл бұрын

Thanks. I am glad you liked my video.

1+1/2+1/4+1/8+1/16+1/32+....=1 I understood it the first time graphically thank you Eugène

@EugeneKhutoryansky

Жыл бұрын

Thanks.

@PunkJackson

Жыл бұрын

Not 1+1/2+..., just 1/2 + 1/4 +.., I know that's what you meant

Here is why Chat GPT has to say about ln2 === The natural logarithm of 2 (ln 2) is an irrational number, which means that it cannot be expressed as a finite decimal or as a fraction of two integers. However, it can be expressed as an infinite series: ln 2 = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... This series converges to ln 2, which means that if we add up enough terms, we can get a very accurate approximation of ln 2. If we want to express ln 2 as a fraction, we can use the first few terms of this series to get a rational approximation. For example, if we use the first two terms, we get: ln 2 ≈ 1 - 1/2 = 1/2 So we can say that ln 2 is approximately equal to 1/2. This is not an exact value, but it is a useful approximation that can be used in many calculations. ln 2 is a transcendental number. A number is said to be transcendental if it is not a root of any non-zero polynomial with rational coefficients. In other words, it cannot be expressed as the solution to any algebraic equation with rational coefficients. It is known that ln 2 is transcendental, as proved by the mathematician Charles Hermite in 1873. This means that there is no algebraic expression or fraction that can represent ln 2 exactly, and its decimal expansion goes on infinitely without repeating. It is not known whether ln 3 is transcendental or not. The question of whether ln 3 (or any other natural logarithm of a rational number) is transcendental is still an open problem in mathematics, which means that it has not been proven or disproven yet. However, it is conjectured that ln 3 is indeed transcendental, based on the fact that most natural logarithms of rational numbers are believed to be transcendental. In fact, it is believed that almost all real numbers are transcendental, in the sense that the set of algebraic numbers (numbers that can be expressed as roots of polynomials with rational coefficients) has Lebesgue measure zero. Despite the lack of a proof, the conjecture that ln 3 is transcendental is widely accepted by mathematicians, and it is assumed to be true in most contexts where such a result is needed. If someone were to prove or disprove that ln 3 (or any other natural logarithm of a rational number) is transcendental, it would be a significant result in the field of mathematics. If the number is proven to be transcendental, it would confirm a widely held conjecture and provide further evidence for the prevalence of transcendental numbers in mathematics. This would be a significant achievement, and the mathematician who proved it would likely receive recognition and praise from the mathematics community. If the number is proven to be algebraic (i.e., not transcendental), it would be a surprise and a significant result as well. It would require new techniques and ideas to be developed, as the current methods used to prove transcendence cannot be used in this case. The mathematician who proves this result would also receive recognition and praise from the community. Regardless of the outcome, the proof or disproof of the transcendence of ln 3 would advance our understanding of the properties of real numbers and contribute to the ongoing development of mathematical knowledge. It is harder to prove that ln 3 is (or isn't) transcendental than ln 2 because these are two different numbers, and each number has its own unique properties and characteristics that affect how easy or difficult it is to prove whether it is transcendental. One reason why it is easier to prove that ln 2 is transcendental is that there are known methods and techniques for proving the transcendence of certain numbers, such as the Gelfond-Schneider theorem. This theorem provides a criterion for determining whether a number of the form a^b, where a and b are algebraic numbers with a ≠ 0,1 and b is irrational, is transcendental or not. By applying this theorem to ln 2, we can prove that it is transcendental. In contrast, there is no known general method for proving that a given number is transcendental. The proof of the transcendence of a specific number often requires the development of new techniques and ideas specific to that number. Additionally, the fact that ln 3 is not known to be transcendental does not necessarily mean that it is harder to prove its transcendence. It may simply be that there has been less effort or attention devoted to proving the transcendence of ln 3 compared to ln 2.

@NL2500

Жыл бұрын

ChatGPT informed me that the lion was the largest mammal. While we all know that an elephant and whale are bigger. ChatGPT is (at the moment) totally unreliable when it comes to facts.

Very nice

@EugeneKhutoryansky

Жыл бұрын

Thanks

What is the title of the musical piece and who was the composer? It’s well known.

@EugeneKhutoryansky

Жыл бұрын

All the music in this video is from the free KZread audio library, and the names of the songs are the following. Renaissance_Castle William_Tell_Overture_by_Rossini

Hey, could you please consider making a video about pararell universes? I would love to hear your take on it, with your animations!

@EugeneKhutoryansky

Жыл бұрын

I have a video on that at kzread.info/dash/bejne/i2iCzdaqdM2_iNY.html

@teemo8247

Жыл бұрын

@@EugeneKhutoryansky Oh right, I've watched that video a couple times already lol

As long as we can get -1/12, we're cool.

Next you could try Ramanujan and Borel summation. 1 + 1 + 1 + 1 + ... = 1/2 1 + 2 + 3 + 4 + ... = -1/12 Then you get into the ideas of regularization and analytic continuation. For example consider the geometric series (valid for |z|

🤯

Would be nice, if you tried to rearrange convergent series to show order doesnt matter

“Riemann series theorem” is never mentioned or explicitly discussed despite being in the title.

@EugeneKhutoryansky

Жыл бұрын

The entire video is about the Riemann series theorem, even though I never mention its name.

Name of song used?

@EugeneKhutoryansky

Жыл бұрын

All the music in this video is from the free KZread audio library, and the names of the songs are the following. Renaissance_Castle William_Tell_Overture_by_Rossini

I am not sure (Jinn) i read a passage from Stephen Hawking book A Brief History of Time, where he mentions Albert Einstein learned from Riemann.. Why this genius hardly ever gets a mention or credit is a Paradox within a Paradox

Please make a video on spherical harmonics!

@EugeneKhutoryansky

Жыл бұрын

That is on my list of topics for future videos. Thanks.

@dl1083

Жыл бұрын

@@EugeneKhutoryansky Thank you!! :)

@sophiophile

Жыл бұрын

​@@EugeneKhutoryansky Please do this, yeah. Especially with the throughline of electron orbitals + periodic table structure.

The mathematical equivalence of making polls say whatever you want 🙂

There can't be an infinite number of fractions of a whole because if there was it'd be impossible to move between any two points, because at any part-way point the remaining distance could always be halved and so you'd never be able to complete that distance because no matter how close you got there'd always be a half distance to complete before the full remaining distance could be completed. That's also true at each part-way point which was already completed. But because we know movement (completing a journey) is not impossible there can't be infinite fractions of distances or spaces.

@sophiophile

Жыл бұрын

Zeno's 'paradox' was solved like a thousand years ago, dude.

@anonimushbosh

Жыл бұрын

@@sophiophile That so called solution is that despite the numbers of half ways between two points being infinite the total of every step between the same two points is finite. That's not a solution that's a restating of the problem. Modern proposals talk about different types of infinities which (like how many numbers there are compared to how many even numbers there are) which just exposes the limitations of mathematicians. A logical solution is that there can't be infinite fractions of a whole.

The inifinity in act can only exist in the mind. Every physical magnitude we know of has a maximum or minimum finite limited value. So this is how we prove the existence of God: A mind that is infinite. And we, made in his imiage, can know Him and the infinite, even if it cannot exist in the physical reality. What can exist in the physical reality of the infinity in potency. But not in act. God is infinite in act.

I hold the opinion that shuffling terms around can't change the true sum as you're just pushing a lot of terms away to infinity and ignoring them. Part of the confusion around this seems to stem from mathematicians being unable to comprehend infinity. Just treat it as an unknown variable of known extreme magnitude.

@EugeneKhutoryansky

Жыл бұрын

The sum really does converge to a different value, and we are not ignoring any of the terms. The mathematical proof is available at en.wikipedia.org/wiki/Riemann_series_theorem

6:17 boss fight

1:54 so the harmonic series isn't absolute which means that the alternating harmonic series has different values if arrange different ?????

@user-pr6ed3ri2k

Жыл бұрын

omg it's showing the harmonic series now

@user-pr6ed3ri2k

Жыл бұрын

ln(2)?yes

@user-pr6ed3ri2k

Жыл бұрын

Thadtslogedts

@user-pr6ed3ri2k

Жыл бұрын

imagine 1+1+1+1++1+1

@user-pr6ed3ri2k

Жыл бұрын

wait so absolute convergence is if sun of absolutes converges?

Original narrator

Great work sir

@EugeneKhutoryansky

Жыл бұрын

Thanks

@TheManOfTomorrow

Жыл бұрын

*mam

@Lanthium-ashu369

Жыл бұрын

@@TheManOfTomorrow i don't know about 😅

@EugeneKhutoryansky

Жыл бұрын

I am a male. The narration is done by my friend.

@EugeneKhutoryansky

Жыл бұрын

Kira Vincent narrates all my videos. This is written on the screen at the end of every video. I live in Houston.

You didnn’t even mention the option of rearranging the terms so that the sum does not approach any real number *or* diverge to ±∞

@EugeneKhutoryansky

Жыл бұрын

Yes, I know. In each video, I have to decide what to include and what not to include. Thanks.

Love a lot the videos, but why repeating the same exact sentence sometimes in them? It always feels odd 😅4:11 4:29

@IlIlIllIlI

Жыл бұрын

I think it is to emphasize the statement

4:20, 4:40... So infinity is -not- a number which can be positive or negative with a vague value used for comparing different infinities and whatever we want it to be. Invented by humans unsatisfied with reality. Infinity is not accepted by the IRS.

* முடிவில்லாத தேடல்கள்*

The universe is computational... This "paradox" exists... Therefore the universe is infinite.

So addition is not associative?!

@EugeneKhutoryansky

5 ай бұрын

If we have an infinite sequence, then not necessarily.

How much money do you make from these vids?

@EugeneKhutoryansky

Жыл бұрын

Not a huge amount. I still have a full time job unrelated to making these videos.