Infinite Sums | Geometric Series | Explained Visually
HTML-код
- Опубликовано: 16 сен 2024
- Geometric series are probably one of the first infinite sums that most of us encountered in high-school. When I first heard of an infinite sum(two or three years ago), I was really amazed that some of them would equal to a finite number. It seemed very strange that even if I keep adding numbers forever I would get a finite answer. At school I was just taught to plug the numbers into the formula, without fully understanding why or how it works.
In this video I go over some examples of geometric series and how we can get some insight on why it works by using visuals.
Thanks for watching~
P.S.
I will be moving to China for 5 months tomorrow, so I'm not sure if I will have an access to RUclips there, due to many sites being blocked by the government.
_________________________________________________________________
Support my animations on:
/ think_twice
_________________________________________________________________
Any further questions or ideas:
Email - thinktwiceask@gmail.com
Twitter - / thinktwice2580
_________________________________________________________________
Programs used:
- Cinema 4D
- Adobe Premiere Pro
_________________________________________________________________
MUSIC:
A time of wonder
• Video
Started binge watching your videos and I feel like crying
Im really glad you enjoy watching my videos
It's because of the music
Noam Tashma it probably has to do with the music, but also the beauty of mathematics and how it's presented
Noam Tashma yes music is an important part of the video. Tbh it takes me a while to find a nice/suitable song for the background.
SAME
Your work is amazing and I would call it "Mathematical poetry in motion". It should be used in all schools to motivate the kids and show them how majestic mathematics really is!
thanks to 3Blue 1Brown for introducing me to this channel !
Bruce In which video does he introduce this channel?
It's a treasure not a channel
Same here
jesus how do you not get more attention? this is beautiful
EonIdiot I guess it just takes time to build an audience. Also I don't really promote my videos in anyway so not many people can find them. Thanks for the comment:) glad you enjoyed
Happy day: I found your channel, thanks 3b1b. The quality is impeccable, people like you should live forever.
Beautiful as always.
Yaman Sanghavi thank you:)
The music along with the animation gives me chills. Good job!
And people say Mathematics isn't an Art.
Its much more....
Who. Said. That
I would say it entearly depends on the artists, and what methods they cuse to preform there arts. What I do not know is "what is not art"? What do you think about this?
Literally nobody has ever said that.
Technically in some cases you need math for art, ʅ（◞‿◟）ʃ
Speechless only tears
Only tears in my eyes....
Thank you
This is the first video I've watched from you.
It. Is. GLORIOUS!
Thank you~
Think Twice Your welcome! ^^
I'm honestly so glad I found your channel
I didn't think it was possible to visualise the formula for the general arithmetic series
Naris R , Same, that was so pretty when it showed up. Reminds me of the Numberphile video on the mandelbrot set from a while back...
Arithmetic series? Wouldn't its sum look like a trapezoid? Its English Wikipedia page got some nice visualizations.
en.wikipedia.org/wiki/Arithmetic_progression
BTW, if you're really into that, Mathologer got an intense video about sums of powers.
ruclips.net/video/fw1kRz83Fj0/видео.html
0:19 standing ovation man for that little animation!
You are seriously talented. The combination of great music and amazing geometry brings me to tears.
Mark Kinnard thank you so much:)
You sir, are a credit to the world - especially to visual learners.
For more stuff like this I can recommend you a book called "proofs without words" it's full of visual proofs like these. I think you will enjoy it:)
I'm using RUclips from 2018. This is the best channel I have ever come across 🥺🥺❤️.... Finally RUclips suggested me some worthy .
Thanks!
@@ThinkTwiceLtu I am thinking twice , think twice replied me 🤔🤔
ahhhhh feel like a child again, love it, so simple yet so elegant.
That last one was BEAUTIFUL
Thanks a ton for this! I was wondering the same about infinite sums and thought that I'd get my answer from a channel like Numberphile, but here you are with a simple, elegant and delightful visual presentation! 10/10! :)
PS: Thanks to you, I've now made it my agenda to apply this methodology to some infinite series that intrigue me personally. I don't have words to describe this newfound enthusiasm!
The best channel ever, there’s no other RUclips channel like you. Keep it up
It’s sad to see how this channel is really underrated.
Thanks for the kind words, I'm glad you enjoy my videos:)
I remember coming up the square visualization on my own when I was first studying power series. Despite being simple it is till date one of my favorites
beatiful visualisations
Math is just beautiful when explained correctly. I've learned and paid more attention to your videos on algorithmic series more than any math related course or explanation I've received before. Truly beautiful.
Proofs without words are the best. Thank you very much.
It breaks my heart to see this channel sooo underrated. I don't usually write comments but you are different. Very beautiful content.
Amazing, I love math, I'm a physics teacher from Portugal. Why this was never shown in books? I had to become 44 to see this, and I first learned when I was still a tennager...
Thank you so much!
Thoroughly enjoyable. The last one took me a bit longer to get but, once I did, I really understood what the geometric series was trying to convey. Very beautiful.
awesome
:)
Amazing! And you can explain the math without saying anything. Its almost like art.
Marcos Lourenço It IS art.
Math is art
I've never felt this enlightened before. I never even believed that an infinete sum can be equal to a number. My friends tried to convince me so many times but it was just way too unintuitive. I need to share your channel with literally everyone I know. Everyone needs to see these videos T_T
One does not *believe* in maths anyway...
A lot of thanks. We need this type of illustrations. Lovely important and helpful.
Whoever came up with the idea of splitting up a triangle into equal pieces over and over again and adding them all up to get a required sum. It's beautiful.
Could you do a video showing the visual proof for the infinite series in the form *(a/n)¹ + (a/n)² + (a/n)³ +...* for *a
Both the animation and the smooth music makes an infinitesimal entity!
I'm in love with whoever made this
Fantastic! It is the beauty of math.
Freaking best videos ever. Appreciate your work, keep it up !!
The beauty of maths. So well done and crafted. Soothing and mesmerising both.
I have never thought about a geometrical interpretation of the common a/(r-1) formula, and it's nice to see it represented on a cartesian plain
This is where Maths blend with Arts, visual and musically. Beautiful. Poetry. Lovely. Thanks
You're an artist
I am blown away!
This channel captures the beauty of math and puts it into a well designed video. Very nice work!
This geometric sum has a property called convergence! This property says that with each added term, we get closer to a final value that is not infinity, for that we can use 2 things, either geometric resolutions (much more creative and cool to see) or a bit of the law of limits present in the calculus
Imma start binge watching your vids now... this is so awesome and I love visually appealing videos like this :))
Ella Puerto thank you:)
This also illustrates why the radius of convergence is 1; the slope of the upper line needs to be smaller for it to cross the lower line
What a beautiful proof! What a beautiful video!
Those numbers and animations are AWESOME!!!!
thanks a lot!
Jesus, i get all shivers
I really was watching for a channel like this, it's awesome
SVP glad you like it~
I've always liked the square example, but the triangular one is cool, too!
These are the most beautiful kind of proofs imo
A very good video, lad to see you back to the usual standard after last weeks mechanaruto. I am also happy to announce that i am the new manager of the think twice channel and if you want to send mr twice think eny memes then you gotta send them my way and if they are spicy enough then he will se dem. TY. skibbydappapp
this is a family friendly channel, please refrain from using such vulgar language
still waiting for the memes
Just beautiful.
i really like this visualization its satisfying
I liked your visual proofs very much ❤
You make me fall in love with math even more.
1:58 greatest geometric based channel in milky way
really helpful illustration. Thanks alot since i feel meaningful of math now
My prime goal before starting my bachelor was to make mathmatics a way of procrastination. Many thanks for helping me get there!!
Mind = blown
I didn't think this could be visualised...
Your videos are amazing! Please don't ever stop!
Ganster 123 thank you~
these days i dont feel too well to make a video. I'm slowly working on a new one but It will take some time. Have a great day~
The music as well! Keep up the good work.
GREAT EXPLANATION!
really intuitive!
thanks!!!
What a brilliant video
Thank you 😊
The last one is called the Mccabe-Thiele Method
In the last expression you can divide both sides by "a" and get a general formula for any value "r"
hated my real analysis course, but I really love this video
this video is my motivation to revise this stuff for exams haha
Another gem. Although I knew the result in advance, the sum of the powers of 1/4 is visually surprising. What will you be doing in China? I think (hope?) you'll have access to RUclips. If not, let us know when you return!
dubarnik thank you. I will be learning Chinese for 5 months there. I will try to access youtube with a help of VPN
For many of these, there is a general case that works in every base greater than one:x = 0.111111111...10x = 1.1111111111...10x - x = 1x(10 - 1) = 1x = 1/(10 - 1)
Consider what happens in base 1. At first x is an infinite sum of ones. But at the end, 1(1-1) = 1/0. That matches the trend of 1/x as x approaches zero, so maybe this is okay.
But if you try to use bases less than one, the results are seemingly paradoxical. For example, in base 0.5, x = 2+4+8+16... at the beginning, and x = -2 at the end.
I quite don't get any of this but I must say it's soothing af.
brilliant
Nice music haaha, you always choose the one that fits!
thanks! it take so long to find a nice song haha
Nice visualisation.
beautiful ! instant like and subscription!
You really deserve more subscribers
Please do a video of trigonometric angle sum identities
Magneficent!
I think you may have just succeeded in making me like maths and number patterns. 😁
Beautiful! Thank you for this.
Im from Argentina;sorry ,my english is poor;but math is music ,this kind moves some inside;and makes me feel near my father and my mother,i hope be for ever with them,soon.
yes bro, i like this videos where Maths is applicated!
i m gonna follow you!! 😀😀👍👍👏👏👏
I didn't understand any of the mæth in this video but it looks cool
This is a very interesting way of learning series and visualizing them.. How do you know what shape to use to solve a series this way?
wow.....awesome man...
This is beautiful. Keep it up!
than you-,very simple, organic, musical
This is beautiful!
My mind has exploded... in a good way.
Wow. Just, thank you.
For the past 3 years I have been clinically depressed knowing molten salt reactors can end poverty and reverse climate change. Watching your videos reminds why life is beautiful.
Beautiful
It would be great if you show the ramanujan's infinite square root of value 3
In a more general case:
Theorem 1.1: ∀a ∈ ℕ*
\frac_{1}{a} = \sum_{i = 1}^{\infty} \frac{1}{(a + 1)^i},
or:
1/a = lim n→∞ ∑ᵢ₌₁ⁿ 1/(a + 1)^i
Hint:
Start by proving that 1/a = 1/(a + 1) + 1/[a(a+1)]
It's not difficult to prove this, it stays as a homework シ.
Completely awesome!!
That was awesome! Thank you so much
The triangle one was awesome
BEAUTY OF MATH U EXPLAINED VERY WELL LIKE HIT SHARE EVERYBODY THIS IS TRUE EDUCATION
when explained in a way that makes sense how, infinity simultaneously is forever and finished, ....all problems will be solved
Took me a second to figure why (1/4)^2, (1/4)^3, ... worked in this geometrical interpretation