Infinite Geometric Series Sum

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This is a short, animated visual proof demonstrating the infinite geometric series formula for any positive ratio r with r less than 1 and with positive first term a. This series is important for many results in calculus, discrete mathematics, and combinatorics.
If you like this content, you can support my work at www.buymeacoffee.com/VisualProofs Thanks!
Also, check out my playlist on geometric sums/series: ru-vid.com/group/PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iaw
This animation is based on a proof by The Viewpoints 2000 group from the October 2001 issue of Mathematics Magazine page 320 (www.jstor.org/stable/2691106 ).
#mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries
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Опубликовано:

29 апр 2024

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Комментарии : 146

@МаксимАндреев-щ7б 5 месяцев назад

This picture is like the picture for the trigonometric proof of Pythagorean theorem

@changchang6806 5 месяцев назад

Yes, the latest and newest one

@RobloxPrompt 5 месяцев назад

Hey I have that exact pattern except its a raster triangle and basically n is the height value for that triangle. (1,n) is basically n for all numbers. and (2,n) would be n%(n+1) excluding 1 from the coordinates for (2,n).

@victorlopezperez3812 5 месяцев назад

That was the same thought I had when I found out about that proof, had to come back and find this

@pqb0 4 месяца назад

well ofcourse it is. The proof is based in a geometric series which converges

@Hamza_Khan_Journey 4 месяца назад

Very neat perspective on the converging geometric series

@US-yw3xi 5 месяцев назад

As the video proceeded, I saw myself learning something new and actually useful. Thank you!

@MathVisualProofs 5 месяцев назад

Great to hear!

@Walczyk 5 месяцев назад

this is not a good visual proof. if this is useful to you and you didn’t know it already then you need to study more

@US-yw3xi 5 месяцев назад

@@Walczyk what do u expect from a 10th grader? -_-

@NewChannel-mm2zi 5 месяцев назад

@@Walczyk Tf do you mean it's not good? A good visual proof doesn't have to help you remember something, it just has to be a cool representation

@ethanciak2517 4 месяца назад

very very elegant

@raymitchell9736 4 месяца назад

Neat proof, I'm trying to recall long ago how we handled converging series... and once again those isosceles triangles are so underrated

@Dr1z. 5 месяцев назад

Thats brilliant. Keep making cool videos like these mate

@vk2ig 4 месяца назад

Excellent proof - thank you!!!!

@Phylaetra 5 месяцев назад

This is a brilliant visual proof! Thank you!

@zix2421 3 месяца назад

Wow, it looks cool

@SkalopSkalop-xo6mj 4 месяца назад

The beauty of the geometric series that you can demonstrated its sum with cool way like this besides the analytic way, and your explanation is good, great video

@mcalkis5771 5 месяцев назад

What a beautiful proof. Thank you for making this.

@MoeinAlizadeh88 3 месяца назад

Great way of proving.

@andrewmartin2321 5 месяцев назад

i love this proof, it’s so easy to derive i literally never will learn the formula except temporarily

@cayoford9486 5 месяцев назад

really cool proof, and I love the way you explained it, minus it going by a bit fast and me needing to pause a couple times to take it in

@MathVisualProofs 5 месяцев назад

Yeah. Hard in 60 seconds. Slower wordless version is linked. :)

@G8tr1522 5 месяцев назад

@@MathVisualProofs it's sad that this is not a standard practice, but i really appreciate you doing that 🙏

@pelayomedina2174 5 месяцев назад

There is a small detail, the lines only intersect if r=1

@G8tr1522 5 месяцев назад

that's what you think. 😈 it's totally valid if it's greater than one, can be proven semi-rigorously in other ways, and has useful applications in physics. You just have to ditch the axioms of convergence. This is our moment in history where we discover fractions, then the number zero, then negative numbers, then irrational numbers...algebra...calculus... complex analysis...then the Reimann Hypothesis! which will (hopefully) advance humanity into a new era of understanding. I think these divergent sums are going to be the key to solving the Reimann Hypothesis....but who knows. 🤷‍♂️ Are you a new student by chance?

@roelin360 5 месяцев назад

@@G8tr1522this isn't the kind of thing you learn until eitther university or consumption of recreational maths content

@G8tr1522 5 месяцев назад

@@roelin360 it's called "inspiring curiosity"

@roelin360 5 месяцев назад

@@G8tr1522 I was referring to you thinking they were a new student.

@IfeelFearForTheVeryLastTime 4 месяца назад

@@G8tr1522 how tf is it valid If r=2 the sum will always increase above 2 so how could it become -1 This is illogical

@aboobakarmohammed1242 5 месяцев назад

Ok now I won't forget that formula again❤

@aidan-ator7844 5 месяцев назад

Genuinely beautiful

@afrika-karibianaestudionan4050 5 месяцев назад

Thanks for this video. It is very useful when introducing the concept of limits.

@sainishanthkoora6576 5 месяцев назад

❤ i love this type of understanding

@RunstarHomer 5 месяцев назад

Awesome proof!

@pleasednut3881 5 месяцев назад

this is incredible

@kinshuksinghania4289 3 месяца назад

Visual proofs are the best

@niom9446 4 месяца назад

Dayum this is great

@pushkardiwan6645 5 месяцев назад

I was just working on geometric sequences form AMC 10 yesterday, and derived a/(1-r). This is a neat proof

@ryankarim7141 5 месяцев назад

Yo you do amc 10? What’s ur aops name mine is aaronjudgeisgoat

@SeanSkyhawk 4 месяца назад

I almost thought this was the new proof I've been waiting for :(

@gtrecc5454 5 месяцев назад

Love this!

@sniper2k364 4 месяца назад

Interesting geometric way to compute powers of r

@ClaytonMurzyn 4 месяца назад

Damn bro, that's sick

@mudetz 5 месяцев назад

Very cool proof

@cccexestarted 4 месяца назад

Just so beautiful 😍

@BrianHii-s5o 5 месяцев назад

thank you

@wahoobear6588 4 месяца назад

Cool Proof i like it 🤘🏻

@sh2157 5 месяцев назад

Brilliant video

@Ninja20704 5 месяцев назад

Its interesting that this proof also shows why we need r1, the lines would get further apart and never interesct. However, I wonder if there is a variation of this proof that could get work for negative r values. That would be very interesting

@MathVisualProofs 5 месяцев назад

It’s on my channel in wordless format from the same cited source. :) I haven’t made it into a short yet.

@kilian8326 5 месяцев назад

They intersect if r>1 but in the third quadrant, and, if im not wrong, with like r=2, it gives -1, as the best method of assigning values to r>1 so it still holds for r>1

@Ninja20704 5 месяцев назад

@@kilian8326 no it doesn’t. If r>1, the terms will get bigger and bigger and the sum will diverge to infinity. Digramatically, you must start with the y-intercept of y=rx+a, then move over to y=x. So you can’t go backwards into the third quadrant unless a is negative. (Which would require a different diagram altogether)

@MathVisualProofs 5 месяцев назад

@@kilian8326 sounds good if you use the 2-adic valuation :)

@HEAVENLY_DEMON_ALPHA 5 месяцев назад

Very nice explanation, keep it up👍

@atcihaydar 4 месяца назад

Amazing

@vikramnagarjuna3549 5 месяцев назад

Stupendously created... Wow video... 🤝🤝🤝

@TheOriginalDeaf 4 месяца назад

This looks like the same triangle for proving Pythagorean theorem using trigonometric functions, everything in math is related in some way or another ❤

@MathVisualProofs 4 месяца назад

And this geometric series proof was also found by high school students.

@nilsalmgren4492 4 месяца назад

Try doing the algebra starting with x= rx+a and solve for x. x-rx = a. x(1-r) = a. x = a/(1-r)

@afrika-karibianaestudionan4050 5 месяцев назад

Greetings from Curaçao, an Island Nation in The Caribbean.

@MathVisualProofs 5 месяцев назад

Welcome!

@G8tr1522 5 месяцев назад

I immediately sniffed out the application to diverging infinite series. So now i have a cool new way to prove why 1+1-1+... = 1/2, but even better, my sleep schedule will be royally screwed up now, bc i am absolutely exploring this rabbit hole. 🤪

@shashu990 5 месяцев назад

Brilliant

@ill4374 5 месяцев назад

amazing

@najmulhasan2484 3 месяца назад

গুণত্তর ধারার অসীমতক সমষ্টি 👍

@sp277 2 месяца назад

Genius!

@luisvasquez5015 5 месяцев назад

This one should be in The Elements 👏👏👏

@Sam91 4 месяца назад

One question... Why did we take the intersection point to be a/(1-r) in the first place

@MathVisualProofs 4 месяца назад

It is the intersection point of the two lines. Set the y-coordinates equal and solve for x if you want to check.

@j10001 5 месяцев назад

That was cool!

@Rosemarytwigs 5 месяцев назад

i dont know anything about math i just like these look

@TomatOgorodow 5 месяцев назад

Short tourist

@castor5580 5 месяцев назад

Thats why its called geometric progression 😊😊

@greggassen5548 5 месяцев назад

My man was doing distillation column design without Even Realizing

@johnpinckney7269 4 месяца назад

cool!

@Fantasticleman 5 месяцев назад

And the same visual proof also works with the vertical segments along the y-axis, right?

@MathVisualProofs 5 месяцев назад

Yes! I just chose the horizontal ones because I like blue better than yellow ;)

@IIHawkGamingII 5 месяцев назад

I just took my calc 2 final yesterday, sequences and series were the only thing I actually didn't understand at all. RU-vid is messing with me by showing me this.

@armchair8258 5 месяцев назад

Makes sense, since if r > 1 or r=1, the lines never meet.

@chrisyo4461 3 месяца назад

Archimedes would be proud😊😊

@_umarro_ 4 месяца назад

im bad at math, can you please explain how you immediately know that the 2nd vertical line equals ar, the third one equals ar^2, so on so forth??

@nachopuma13 5 месяцев назад

Nice to see how if r>=1 this doesnt work as the lines never intersect. The sum diverges

@Mathisgood4u 4 месяца назад

Bro, what the sigma? I’m only a brainless shorts viewer, I can’t be watching this, it’s too much for my brain, wait, do I have one?

@unrealredist4124 4 месяца назад

I'll help you!!! *Skibidi*

@dariusvascuta8024 4 месяца назад

sooo coool

@vennstudios9885 5 месяцев назад

so the product of A and the Sum of all r Notations from 0 to ∞ is equal to A over the difference of 1 and r this is some fractal stuff right here

@anilkumarsharma8901 5 месяцев назад

Means simple lines are results of some geometric series which have some inclination????

@megamuslimchad 4 месяца назад

That’s cool.

@heeraksharma1224 5 месяцев назад

Wow.

@jellyfrancis 5 месяцев назад

r>1: it diverges Complex analysis: wait, hold on....

@hiredfiredtired 5 месяцев назад

1+2+3... = -1/12

@jellyfrancis 5 месяцев назад

@@hiredfiredtired exactly

@hananebouattour8870 5 месяцев назад

Can you just make a similar one with the same principe but try the intersection of y=x² and y1=x+b

@angelcosta4383 5 месяцев назад

Or alternatively, divide by a and you get 1+r+r²+r³+r⁴+...=1/(1-r)

@sochex 5 месяцев назад

Where r

@johnmillerpere_grin6371 5 месяцев назад

Basically, my statement stands true. Whenever you get infinity involved in mathematics, you come to the existential crisis of there being no such thing as any real number. We only have definition for anything because intelligent beings define them.

@Machhi_Pranay 5 месяцев назад

Could You Do same proof taking as r >1 ? 😁 so You have to take slope > 1 meant Angle with X axis > 45°

@Ninja20704 5 месяцев назад

If r>1 then the sum diverges and does not give us any value. You can see that digramatically because if r>1, the two lines would get further and further apart as x increases, never intersecting.

@Machhi_Pranay 5 месяцев назад

@@Ninja20704 Lets think with this video's way, What if for r>1 we take the line y = rx - a (GP : first term -a , common ratio r) How would be Result? 🤔😁

@Ninja20704 5 месяцев назад

@@Machhi_Pranay in that case you would have to draw your diagram in the third quadrant because all the values will be negative. And as you go further left you see the same thing of the two lines getting further apart, showing that the sum diverges towards -inf

@d1scocubes 9 дней назад

Nice work but what if I break your rules?

@yanben 5 месяцев назад

Holy what the heck insta sub from me

@MathVisualProofs 5 месяцев назад

Thanks!

@DanielPoupko 4 месяца назад

And also that's the fixed point.

@PC_Simo 4 месяца назад

I’m here struggling with understanding, why the intersection point should always be at x = a / (1 - r). Since y = x, at any point, on the graph of y = x; that means that, if I substitute a / (1 - r) for x, in the function rx + a, I’d get: a / (1 - r), as the output. But I keep getting: ar / (r - r²) + a. 😅

@MathVisualProofs 4 месяца назад

when you plug in x=a/(1-r), you get ar/(1-r) + a, which is ar/(1-r) +a(1-r)/(1-r), adding the numerators gives a/(1-r)

@PC_Simo 4 месяца назад

@@MathVisualProofs That’s right! For some reason, I also multiplied the denominator 🤦🏼‍♂️. Maybe I should sleep, every now and then, lol 😅. Anyway; thanks, for the quick and clear answer. Have an awesome day/night. 👍🏻

@MathVisualProofs 4 месяца назад

@@PC_Simo happy to help. easy to make mistakes like that for sure.

@PC_Simo 4 месяца назад

@@MathVisualProofs Yep. On the Numberphile-video: ”Epic Circles”, I gave some bogus formula, for a general fraction in the sequence, that was actually linear, when it should have been quadratic. 😅

@anilkumarsharma8901 5 месяцев назад

What is maximum to minimum value for r and x

@deananderson7714 4 месяца назад

Think about what would happen to this proof if r was 1 or higher. If we have r=1 then the slopes are equal and the lines are parallel meaning they won’t cross making this proof break down. Similarly if r is greater than 1 the slope will be larger than the first line and they will instead intersect in the third quadrant not the first again making this proof break down. Similarly for negative r this triangle isn’t formed which is why this proof only works for 0

@ralvarezb78 5 месяцев назад

a fractal

@f.r.y5857 5 месяцев назад

Only if |r| < 1

@alanspyglass3304 5 месяцев назад

this is the new one right? Those highschoolers found it a few years ago no?

@MathVisualProofs 5 месяцев назад

Video has linked source - year 2000. But it does have a similar look to the more recent new proof of Pythagorean theorem.

@Liam-bh8qc 4 месяца назад

WHY IS NO ONE RELISING? bruh.

@dunkadmaklaud 5 месяцев назад

Freddy Fasbear: ar, ar, ar,...

@Liam-bh8qc 4 месяца назад

It DID NOT FILL COMPLETELY ALL THE WAY. WHY DOES NO ONE KNOW?

@НиколайШерстюк-ы7е 5 месяцев назад

Literal geomertic series 😆

@amineaghbal3 5 месяцев назад

We can use Taylor serie to prove it

@shreyashkhandelwal5861 5 месяцев назад

I didn't understand it please explain it to me 🗿

@Joe_Payne 5 месяцев назад

Cool story. This was actually discovered by a group of school kids and is special because it doesn't require many rules like a lot of other proofs.

@antigay8346 5 месяцев назад

0:02

@rcb3921 5 месяцев назад

dang

@ythegamerita 5 месяцев назад

I Will pretend like i understand what you said

@あかし中村 5 месяцев назад

Fuck 🤯 It's that easy?

@mircoceccarelli6689 4 месяца назад

👍👍👍😁🤪👋

@dougr.2398 5 месяцев назад

Slow down. This is with the extra time

@MathVisualProofs 5 месяцев назад

For a short format, this is all I can fit. The linked video goes slower (but is wordless).

@DarkInTheNight 4 месяца назад

Iterative methods for rootfinding…. This is just defining the geometric series recursively

@colinjava8447 4 месяца назад

That's pretty neat, but in higher level maths they prefer algebraic proofs over visual ones, similar to venn diagrams, you need algebraic proof, but the diagram can be used for intuition.

@typhon8047 5 месяцев назад

@Walczyk 5 месяцев назад

you don’t justify that’s the triangle is isosceles at all. lame

@hiredfiredtired 5 месяцев назад

mate, it shares 2 of the same sides. What other justification do you need

@enkor9591 5 месяцев назад

You know two angles, 90° and 45° (that's the angle of the y=x graph)

@edwardhuff4727 4 месяца назад

@gperm4941 You beg the question. The horizontal line being 𝑎 is justified by claiming that the triangle is isosceles. To prove the triangle isosceles (two equal sides), note that it is a right triangle and one angle is half a right angle, and the angles of a triangle sum to two right angles, so the other angle is also half a right angle, and a triangle with two equal angles must be isosceles.