Alternating series #1 | Visual solution |

[@ThinkTwiceLtu]

Hello everyone. I just wanted to say thanks for all the support I've been getting from you guys. Means a lot. P.s. I will be starting my undergraduate degree in a few weeks so It might affect my upload schedule. Anyway I'll keep you guys updated on twitter. Peace~

[@Kyle-li8wi]

Undergraduate?!?!

[@jokerpen15]

undergrad degree wtf?? u must be a genius

[@ThinkTwiceLtu]

I'm just a guy with an animation software and an access to the internet. Whenever I read about some topic that fascinates me I just want to share it with you guys so I make these animations:)

[@maaaaaaaaarcel]

What sources on the internet do you use?

[@AnubhavGoswami3800]

You're making some amazing stuff, man. An inspiration, really.

[@james3742]

Probally one of the most intuitive representations of an infinite geometric series that I have ever seen. Thank you for everything you do.

[@nethols2185]

All of these help so much more than actual words

[@WhiteboardMaths]

These animations are incredible!

[@yoavzack]

this is just brilliance in action

[@OrangeC7]

These visual proofs are amazing, never stop making them if you can help it!

[@hubriss583]

I love your animations so much! They make maths less obscure

[@vpambs1pt]

I find very interesting that many series and be "intuitively" seen, once you find the "perfect" scenario for each series, but the real problem is find the perfect scenario, for example who'd have though in solving this series geometrically, by using 3/4 of a square! By the way, I love the fact that in title of the video, I can see a "#1"!

[@ThinkTwiceLtu]

yes I agree:)

[@thearmadilliestone]

Series blew my mind when I first encountered them in calc

[@daksh8747]

Same

[@italyball2166]

Elegant as always!

[@ThinkTwiceLtu]

thank you~

[@Udinanon]

Majestic

[@apofistetrahidrocannabinol3949]

Beautiful, as always

[@ethangilchrist3534]

Beautiful as always

[@manuelhoffmann4470]

Very beautiful animation! I have never seen this approach and it blew my mind!

[@WarheaddVids]

Got right to the point and super easy to understand. Great video!

[@thanostitan.infinity]

Thanks for these. All the best for future!

[@ThinkTwiceLtu]

thank you:)

[@Conkerbrain]

Damn, absolutely love your stuff

[@ThinkTwiceLtu]

[@KnThSelf2ThSelfBTrue]

Oh my god. Your content is just... so on it. Why do we put kids through years of grueling classroom math lectures when the answers are sitting right there on KZbin, laid out so clearly, and set to Chill Lo-Fi Beats to Study/Relax to.

[@neo_tsz]

This is phenomenal

Awesome!

[@BrickfilmMan]

Amazing!

[@funkysagancat3295]

Amazing as always

[@nikolamarijanovic6261]

It's amazing how all of the numbers are powers of 2 but somehow in the end we get 2/3

[@Wild4lon]

No. It math. c:

[@TJStellmach]

Consider the decimal 0.111111111... All of the individual digits are powers of 10, but the sum is 1/9th. This sum is similar, except in base 2 rather than base 10. 2/3 in binary is 0.10101010...

[@xavier2.044]

Yay I learned something!

[@chanyoonjun]

Beautiful

[@guestimator121]

Simply beautiful

[@rreirah]

Super cool again m8 cheers up !

[@patrickwienhoft7987]

Often when one of your videos starts I know the formula/result but I don't see how your approach is getting there. The feeling when it "clicks" in my head and I can foresee your punchline (0:34 here) is simply amazing in every single one of your videos. Thanks for your great content.

[@ThinkTwiceLtu]

Thank you for the support:)

[@cabra500]

The simpler ones are so amazing!

[@radoslavhabarda1180]

your videos are awesome

[@yuvaleeus5eva]

I love your videos so much, stay awesome! Like god damn I get such huge nerdgasms watching your videos haha

[@ThinkTwiceLtu]

thank you

[@markthrekrain8037]

Beautiful, as always. Thank you very much

[@linkspring1287]

Wow..😍😍

[@Reforitor]

You're almost at 50k. Best wishes

[@ThinkTwiceLtu]

[@mohammedal-haddad2652]

Ingenious!

[@realcygnus]

cool

[@connor5890]

Woah, that was cool

[@aks9545]

Your editing skill are insane

[@b.a.r.c.l.a.y9701]

God has joined the game

[@joske7804]

great animation

[@ThinkTwiceLtu]

thank you!

[@Inyrth]

Always amazed when watching your video's

[@ThinkTwiceLtu]

[@luizfelipegarcia4676]

That was great!

[@shreyasraut6224]

beautiful... Elegant

[@raterix2]

Nice.

[@JWentu]

lovely!

[@aaronhamilton8997]

Dayyumn. That was good. Give us moarr!!!!

[@yaeldillies]

My way to visualize this sum is by remarking that all negative terms are all half of the positive terms. So, the serie in fact equals 1/2(1 + 1/4 + 1/16 + ...) = 1/2 * 4/3

[@pursuitsoflife.6119]

Yaël Dillies weird question to ask but are you from Martinique?

[@EvilCherry3]

Great work !

[@ThinkTwiceLtu]

thank you very much

[@mmukulkhedekar4752]

Wow this is awesome .....I want some more pls

[@LlNDEN]

This is so good

[@LimeEffy]

Fantastic visualization! What program are you using for the animation?

[@ThinkTwiceLtu]

I'm using Cinema4D

[@rosettemasilang9749]

i really appreciate this video more knowledge 👍

[@santonusharma3899]

You are making amazing stuff ... keep it up

[@ThinkTwiceLtu]

Thanks for the support!

[@alonelyspoon]

Im so happy i found this channel from 3b1b :)

[@ThinkTwiceLtu]

glad you enjoy it:)

[@ethanw2450]

This is so nice

[@Makasu]

is that an "spirit away" remix?

[@ThinkTwiceLtu]

yes~

[@Makasu]

and very cool video!

[@sirknight4981]

*a Tip: Use "an" when the following word starts with a vowel (a, e, i, o, u) and "a" when the folowing word starts with a constanant (a letter that's not a vowel)

[@Makasu]

sorry, I'm not english basically...

[@abhinavshripad5356]

True genius

[@itsok8003]

👍

[@biswajitmahanta500]

Wow no dislikes. Keep it up😀😀

[@ethanjensen661]

That was great

[@Gabtube252]

great vid, would definitely recommend to a Calculus student that struggles with sums

[@trobin]

these animations are fascinating, what software do you use?

[@ThinkTwiceLtu]

thank you! I used cinema 4D for this one

[@christiansinger2497]

Splendid

[@Викторина-ъ3т]

wooow its so cool

[@JustMe-ui9bv]

This is fucking amazing! Can't get enought of this videos! Keep going. You have to.

[@ErojFeeding]

beautiful videos, nice work

[@madhavjha5289]

Amazing

[@ratherblue140]

Woah...

[@ジャナ-b1p]

they should be teaching math this way in schools

[@RiderOfMooses]

What do you use to animate these? Do you code them?

[@redsalmon9966]

I remember this 4 L shape thingy, but I didn’t know it can perfectly visualise this alternating series

[@lucianorodriguez7726]

wow I LOVE IT

[@ThinkTwiceLtu]

thank you :)

[@lucianorodriguez7726]

Think Twice Thanks to you. you make that many people who dont understand or dont care about mathematics are surprised, to my that love maths, I am surprised and I cant stop showing these videos to people, although I know they will never like them as much as I do. Total thanks :)

[@yashuppot3214]

I derived this myself except for 1/2-1/4+1/8 using an equilateral triangle and going back and forth along its sidelength by factors of what ever the term was, it a point along the same horizontal line as the centre of the triangle, which is 1 third up the side length. Therefore it is equal to 1/3

[@Defeshh]

lovely

[@VaibhavKarve]

Great video! A small suggestion: think about typesetting your math equations in LaTeX. They'll look prettier.

[@ThinkTwiceLtu]

true, I'll have to learn the LaTeX syntax at some point

[@undergroundmonorail]

Gorgeous...

[@ThinkTwiceLtu]

thanks!

[@serock3]

Daym, neat

[@Gamikane]

I really should be sleeping right now but I can't stop watching

[@hybrid6522]

mind = blown, have a like duce 👍

[@LinusL170]

Mind = blown

[@nazishahmad1337]

can you do an animation on 1+2+3+4+............... =-1/12

[@non-inertialobserver946]

That's impossible because IT ISN'T - 1/12, IT'S INFINITY.

[@non-inertialobserver946]

Watch this kzbin.info/www/bejne/j6asep2Cp5upi6M

[@xCorvus7x]

ViperDaniel Chill, the mathematical ways to arrive at that result such as Ramanujan summation are a thing and, albeit counterintuitive, might be visualisable. Does that not sound interesting?

[@non-inertialobserver946]

I'd be very surprised if there was a way.

[@non-inertialobserver946]

Btw I know a way of visualizing 1+2+4+8+16...=-1 and other divergent geometric series

[@cavver3523]

Wooow

[@namannarang4208]

Nice vedio man I never expected the answer to be 2/3 I guessed 3/4 or I thought the denominator had to be even

[@jeanhealey1354]

Math porn was created here. The. best. videos. ever.

[@imelse]

Smart smart, nėra pakankamai ilgo gabalo, tai tiesiog padalinai vieną video į kelis :D

[@ThinkTwiceLtu]

haha jo, cia Lauris?

[@imelse]

yup :D

[@ThinkTwiceLtu]

as to ilgo video net neikeliau, sita trumpa greit padariau vietoj to:D

[@imelse]

aaj, bet kelsi ir tą, ane? Aj, ir nice outro.

[@ThinkTwiceLtu]

cj nekelsiu kolkas. nelabai gerai atordo :D. Dekui

[@tranl1050]

Spirited away????

[@ThinkTwiceLtu]

yup:)

[@taharahmani1547]

is it the right way to understand science : visualising in very particular dimensions theorems . you guys have reminded me of the Einstein's approach; trying to understand concepts by imagining thought experiments and finding physical interpretations of fundamental science. that is extraordinary . still is there a remarkable opinions about the nature of mathematics and it's concerns : Henri Poincarré said that one mathematical object can express a non-finite and different phenomena's and as as a consequence it can have an infinite interpretations . as I shall take an example of what I am talking about let us consider the Pythagorean theorem not from the geometrical view but from the one of probability and statistics: we have a pocket which contains "a" balls with different colours ("a" is whichever number we choose). Now, we can only pick 2 balls successively with reset . the number of possibilities ,of combination according to Pythagoras is equal to the number of possibilities if we use another pocket which contains "b" balls of different colours plus the number of possibilities in a pocket of "c" balls . and all of that done in the same conditions of experiment (successively and with reset) . and finally we have our new interpretation witch totally different from the one with cubes (geometric ) . a²=b²+c²

[@natasadjurdjevic3533]

I dont understand why are we starting with 3/4 of a square

[@ThinkTwiceLtu]

you can start with any shape you want. In this case if we treat the area of 3/4 of a square as 1 and cut it up into 4 equal pieces it works out nicely. It's all about finding a suitable shape to start with.

[@EvilCherry3]

The (very good) idea of starting with 3/4 of a square comes from the fact that we already know that the result will be 2/3 before trying to create this visual demonstration.

[@andrewclarke598]

Think Twice it would have made way more sense to me to start with one whole square. The first three terms just get you to 3/4, since w minus 1/2 + 1/4 = 3/4. I honestly found this video confusing because of that.

[@andrewclarke598]

W should be 1 in the above comment

[@MrRyanroberson1]

Well as the sum from 1 to infinity of 1/x^n is 1/(x-1), then sum(1/x^n) for x=-2 would be -1/3. The alternating series is secretly not an alternating series. Amazingly enough the 1/(x-1) deal works for all ||x||

[@taijusakai7276]

Can u do visual proof of 1-1+1-1+1-1+1-1+1-1+1-1+.... =1/2

[@mithat9398]

Oh my

[@snowman7514]

Me : *amezed noise* Brotha : What? Me : Just Sigma(see my) favorite KZbinr's video

[@longplays2420]

The way I did it was to split it into 2 different convergent series. The answer is the same, it just feels cooler.

[@rainsleeper4680]

What does the first initial corner piece being removed represent?

[@srpenguinbr]

Do a visual proof that the volume of a pyramid is a third of the prism with the same base.

[@natnatschool3538]

Since you said, you just finished school: what did it feel like to be recommended by 3blue1brown? (Ps awesome video as always! Really appreciate that he recommended you)

[@ThinkTwiceLtu]

I was really happy of course! I've been a fan of 3b1b for a long time so I was excited when I found out that he will feature me on his channel. It's an honor to me :)

[@Magnasium038]

That is a beautiful start; to use 3/4 of a square. However, I don't think you needed to show the full square first. In fact, that actually confused me; because I thought going from the full to 3/4 square was your first step, and I paused to ponder why you had done that.

[@rohansonawane7634]

Why you have taken a cube of part 3/4 and not total square

[@arijitchakrabarty1552]

Say, if n=1, it becomes 1-½ = ½. So how is your answer correct ?

[@Yu123456Ji]

Arijit Chakraborty you change the value of n, so it becomes (-1/2)^n => (-1/2)^1 = -1/2. You inverted -1/2 wirh n, be careful with these.

[@arijitchakrabarty1552]

No no, that's not what i meant, and also, my question itself is kinda wrong, which i just figured out. Anyways, thanks for the tip.

[@Pedritox0953]

Is the right direction but could more awesome