I heard 1/2+1/3+1/4+... goes to infinity but I didn't think it would go past 3. Reddit r/learnmath

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Күн бұрын

Why does the harmonic series 1/2+1/3+1/4+1/5+... diverge to infinity? Answering calculus questions on Reddit. See the original post on r/learnmath: / why_does_this_sequence...
Get your indeterminate cat t-shirt: 👉 amzn.to/3qBeuw6
0:00 Why does the harmonic series diverge?
1:41 The classic proof
6:00 Integral test proof
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/ blackpenredpen
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#calculus #bprpcalculus #apcalculus #tutorial #math

Пікірлер: 853

@bprpcalculusbasics 4 ай бұрын

"If you add infinitely many positive number you get positive infinity"? In fact, it’s not! kzfaq.info/get/bejne/ebWVjKaqlsrPfaM.html

@Bla_bla_blablatron 4 ай бұрын

is this what we suffered for? To be held down by systematic racism! We Democracy. We vote now. Gibs for all Under privileged inner city people!

@Bla_bla_blablatron 4 ай бұрын

...and sheeeeet

@netdragon256 4 ай бұрын

I didn't watch this, but it's easy with calculus. The integral of 1/x is ln |x| + c which goes to infinity, as x goes to infinity, albeit very slowly. Sure, your summation formula is integers, and the integral is more granular, but it doesn't matter because the CARDINALITY IS THE SAME as you approach infinity! One just grows faster than the other. Obviously, this isn't as pleasing because it depends on the understanding of the cardinalities of the sets are identical, however we know this to be the case.

@Zorgdub 27 күн бұрын

That's true if you're adding integers :P

@VeteranVandal 5 ай бұрын

It's just slow, but it goes "to infinity" through sheer fraction will.

@hardboiledegg2681 4 ай бұрын

the sheer fraction will when I introduce 1/n^2

@obiwancannoli1920 4 ай бұрын

The indomitable fraction spirit

@Saber1320 4 ай бұрын

@@hardboiledegg2681 well doesnt this just converge to pi^2/6 ?

@hach1koko 4 ай бұрын

@@Saber1320 yeah that's the point, this one would converge despite "sheer fraction will"

@jayi31 4 ай бұрын

@@hach1kokothe futile fractional will vs the unstoppable power of a square

@vladimirpotrosky7855 5 ай бұрын

Marker switches so smooth

@kaustubhgupta168 5 ай бұрын

yea hes very good at it

@bprpcalculusbasics 5 ай бұрын

Thanks!

@AdelaeR 5 ай бұрын

I ignored the math completely and was just mesmerized by his sleight of hand color changes.

@diggoran 4 ай бұрын

And now you know why the channel is named bprp, because it stands for black pen red pen. The switching is part of the branding.

@AdelaeR 4 ай бұрын

@@diggoran ***mind blown***

@trillionbones89 4 ай бұрын

Answer: It approaches infinity incredibly slowly.

@EhabShahid 3 ай бұрын

*infinitely slowly

@bohanxu6125 2 ай бұрын

Logarithm: "hold my beer" ... "It's going to take awhile"

@chickenlittle8158 2 ай бұрын

... can things approach infinity any slower or faster than something else?

@derblaue Ай бұрын

@@chickenlittle8158Actually yes. If we take the partial sum and compare them for different series we can see the different "divergence speed" by the ratio of the partial sums as we increase the terms of the partial sums.

@burgundyhome7492 Ай бұрын

Infinite numbers, just like finite ones, are not equal and comes with infinitely different sizes.

@crclayton 4 ай бұрын

I haven't done math proofs or calc. since college but KZfaq has correctly started promoting this content to me and it's really nice to see these clean demonstrations.

@KCJbomberFTW 4 ай бұрын

Same I’m an accountant I got into school with this 😅

@Bla_bla_blablatron 4 ай бұрын

math is Racist! Y'all betta go back to da Crackerbarrel

@pedrogarcia8706 4 ай бұрын

probably means you share interests with college calc students!

@saravanarajeswaran2626 5 ай бұрын

Can you teach a proof why pi is irrational, like assuming it to be rational and prove it with contradiction

@Qermaq 5 ай бұрын

Mathologer has done a really good video on that. It's quite a bit more challenging than this problem.

@maxgoldman8903 5 ай бұрын

kzfaq.info/get/bejne/mqx7lbmFrbGYpIk.htmlsi=n08QZRrYRXzSRdfM

@customlol7890 5 ай бұрын

That exercise is trivial and left as an exercise for the reader - Fermat probably

@maxgoldman8903 5 ай бұрын

kzfaq.info/get/bejne/mqx7lbmFrbGYpIk.htmlsi=01bxPVz8EVOpP7E-

@saravanarajeswaran2626 5 ай бұрын

@@Qermaq Yep i saw in mindyourdecision too, but they're seen to be complicated (even though he showed that easiest proof), so i asked him to do, because it could be easy to see

@worldnotworld 5 ай бұрын

This proof is due to Nicole Oresme, a medieval mathematician, theologian, and philosopher.

@nchappy16 4 ай бұрын

mfs really just had “smart” as their job and worked in every educated field

@worldnotworld 4 ай бұрын

@@nchappy16 So true. And the fields weren't as distinct from each other then.

@A38 4 ай бұрын

@@nchappy16People have that job today. There are even people whose job is just to communicate breakthroughs in science and math to the layman. The difference between the most and least educated humans is EXTREME.

@bardsamok9221 4 ай бұрын

Would be good if he proved God

@dovallebr 4 ай бұрын

@@bardsamok9221Saint Thomas Aquinas already did that tho

@user-nd7rg5er5g 5 ай бұрын

I feel like I'm a toddler being successfully convinced to enjoy eating vegetables by a knowledgeable older person. Math normally slides off my brain like water off the back of a duck, but this explanation style was excellent, and taught me a lot!

@TheAlienPoison 4 ай бұрын

This comment is misleading, I still don't enjoy vegetables after watching the video.

@richardkurniawan6066 4 ай бұрын

she toddlers on my vegetable til i math

@thecrazygamertarun5265 4 ай бұрын

Goo goo gaa gaa daada baabaa

@Matt-lv1jl 4 ай бұрын

Bruh how can one not enjoy vegetables, that stuff is addictive af@@TheAlienPoison

@fernando4959 4 ай бұрын

@@Matt-lv1jl dog what veggie you consuming

@Misteribel 5 ай бұрын

The beauty of this series is that it doesn't work in 3d: the paradox of Gabriel's horn. If you turn the graph around is x-axis to get a nice horn, the total content of the horn is NOT infinity, but converges to π.

@vanshamb 5 ай бұрын

π and e are omnipresent man

@galoomba5559 5 ай бұрын

@@vanshamb There's circles involved. Not that weird that pi appears.

@vanshamb 5 ай бұрын

@@galoomba5559 ohkk then just e is omnipresent ig

@CliffSedge-nu5fv 5 ай бұрын

Infinite surface area, but finite volume. You could fill it with paint, but can't paint its sides.

@alinasar6192 4 ай бұрын

@@CliffSedge-nu5fv daym that makes no sense, maths wrong

@matta5749 5 ай бұрын

Fun fact: even if you only have inverses of primes (I.e., 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + … ) it still diverges to infinity. But if you take out all denominators that contain a 9 from the normal harmonic series (I.e., 1/2 + 1/3 + 1/4 + … + 1/8 + 1/10 + … + 1/18 + 1/20 + … ) it converges to a finite sum. (Credit to that one mathologer video)

@matc241 5 ай бұрын

Lol, that second one seems counterintuitive but actually makes sense thinking about it.

@sdfsdfhgcvbn2425 5 ай бұрын

@@matc241why would the second make sense?

@RCmies 5 ай бұрын

I don't understand

@theeraphatsunthornwit6266 5 ай бұрын

How come the second one converge. . DOES NOT make sense at all. I dont believe it. If that one converge, then the took out part must also be converge coz it is smaller , then recombine them , must result in converge value.

@timonbubnic322 5 ай бұрын

@@theeraphatsunthornwit6266 i agree yeah, it also doesnt make sense considering there are more multiples of 9 than there are primes...

@Joefrenomics 5 ай бұрын

Ah, darn it! I tried to prove it myself first. Failed. Then saw your proof and thought “Darn it! Why didn’t I think of that?!”

@bprpcalculusbasics 5 ай бұрын

I didn't come up with these proofs myself. I learned them when I was a student. I most likely wouldn't be able to come up with them myself.

@MikehMike01 5 ай бұрын

chocolate pudding

@rhel373 25 күн бұрын

@@bprpcalculusbasics My maths knowledge doesn't go beyond... I guess high school level, and I don't really have any clue how a proper mathematical proof works. To me, before having finished the video, it seems like any sequence where the number added doesn't eventually become zero should approach infinity. There's always *something* added so the number will keep growing for infinitely many steps. Given how simple that sounds I can only imagine I'm making a mistake in my logic there, because usually these things are more complex my intuition tells me.

@tjaschicksalVegita 17 күн бұрын

@@rhel373 If you take the sum of 1/(2^n) = 1/2 + 1/4 + 1/8 + ... It approaches 1 not infinity, even though a number > 0 is added to it infinitely times

@eduardoteixeira869 5 ай бұрын

Thank you. I knew this is a divergent series, but I did not remember if I saw a demonstration for that. Thank you to show me such elegant proof.

@tomkelley4119 4 ай бұрын

The first proof that you showed is really beautiful! Thank you!

@thezanycat 3 ай бұрын

Haven’t seen the second integral proof before - really like it! Great explanation

@aneeshbro 5 ай бұрын

Keep up the good content man. You are very regular, and I like that!

@customlol7890 5 ай бұрын

Ayo the pfp?

@aneeshbro 5 ай бұрын

@@customlol7890 coincidence

@Bla_bla_blablatron 4 ай бұрын

you #1 Racist

@CesarDainezi 5 ай бұрын

I saw this demonstration in my Real Analysis class 10 years ago. Very elegant.

@cbunix23 5 ай бұрын

I saw that proof in middle school.

@CesarDainezi 5 ай бұрын

@@cbunix23 In middle school I didn't now what x meant LOL

@zigzagnemesist5074 4 ай бұрын

@@cbunix23You must live in some advanced alien civilisation if you’re doing these proofs in middle school…

@cbunix23 4 ай бұрын

@@zigzagnemesist5074 Haha. It was a long time ago; it might have been first year high school. Heck, my dad did spherical trigonometry in high school 100 years ago.

@anandasatria7734 4 ай бұрын

@@cbunix23Sheet, what kind of alien civilization taught spherical trigs in high school even before ww2?

@EastndFTball 4 ай бұрын

Sweet video dude, thanks for sharing. That was cool and informative

@mechbfp3219 5 ай бұрын

Even getting to a million would take an incredibly large number.

@Hypernova7777 5 ай бұрын

Because the amount for each next ½ doubles it would take 2^1999999 for the *final ½ alone*

@Ninja20704 5 ай бұрын

In fact, the harmonic series grows very similar to the natural log. For sufficiently large n, 1+1/2+1/3+…+1/n ≈ ln(n)+gamma where gamma=0.5772… is the euler-mascheroni constant. So the amount of terms needed to pass a given number N is approximately e^N

@matheusjahnke8643 5 ай бұрын

@@Ninja20704 furthermore... We can approach ln(k), whenever k>1 as 1/n + 1/(n+1) + .... + 1/[kn] as n -> infinity; Round kn in the way you feel best if it isn't an integer;

@ishansh0077 5 ай бұрын

'e' is omnipresent

@GarrettRoyce 5 ай бұрын

@@Ninja20704 I ran it through WolframAlpha just to see what kind of numbers we're looking at here... e^1,000,000 ≈ 3 × 10^434,294 and the series from 2 to 3 × 10^434,294 is ≈ 999,999.5

@SteveThePster 4 ай бұрын

Nice vid. Maybe worth mentioning that the numerical proof is much more powerful as it doesn't rely on the huge amount of calculus behind the theorum/proof for Integral (1/x) = log x

@Bla_bla_blablatron 4 ай бұрын

Y'all stole that from Africa! Or maybe you never heard of Wakanda.

@snorman1911 4 ай бұрын

@@Bla_bla_blablatron we wuz kangs

@operator8014 4 ай бұрын

I remember seeing a neat proof my calc 2 prof showed us, but I don't remember what it was. I think he calculated the rate of shrinkage of the terms, and compared it to the rate of expansion of the sum, and showed that it remained positive for any given number of terms, therefore it must go to infinity.

@matthijssmulders3432 5 ай бұрын

Another amazing video as always

@trrrippytaka 4 ай бұрын

Thank you. I had not seen this proof represented graphically before. Now it makes sense.

@jeffolson4803 4 ай бұрын

This was a great explanation!

@kayleawilson 4 ай бұрын

Informative! Also I love your voice so much

@danielkalafat9501 4 ай бұрын

Love it man. Great proof.

@killer100897 4 ай бұрын

I didn't noticed his second marker at first and got confused on how the color suddenly changed without putting down his arm 😂

@digitig 3 ай бұрын

You didn't know what the channel name, bprp, stood for, did you?

@killer100897 3 ай бұрын

@@digitig nope just saw this video in my recommendation and decide to click it on a whim

@Qcattee 17 күн бұрын

A very good and clear to follow video but also can we give credit to how flawlessly and quickly the switch between black and red marker is! I’m very impressed!

@Embashe 4 ай бұрын

One of the coolest proofs I’ve seen, love it - thanks!

@eoinmaher8916 4 ай бұрын

Great video, very well explained

@spencergrover6886 4 ай бұрын

Learned this in calc class yesterday. I think my phone is listening into my classes.

@indigoziona 4 ай бұрын

You somehow made it sound simple *and* made me feel smarter just watching 😊

@superuser8636 3 ай бұрын

Shout out the limit comparison test and integral test (in order according to powering). This video is literally amazing

@coldandafraid 5 ай бұрын

That geometry proof is nice

@LetalisLatrodectus 4 ай бұрын

Awesome video; very clever proof

@robertmayster7863 4 ай бұрын

You, Sir, have my upvote

@alexpison628 2 ай бұрын

so fresh and so clean🎉

@getsajal 4 ай бұрын

Wow. This was a great explanation.

@DenisTruong 4 ай бұрын

Wow great explanation ty

@blarblablarblar 4 ай бұрын

Cool. I was thinking something involving prime numbers where you could also represent the sum as a sum of all series for every prime where there are an infinite number of primes or something, but this is nice

@ShabDhdgdh 4 ай бұрын

this dude's got me watching a math video for entertainment

@raimundo116 4 ай бұрын

I just smiled after I got it, when he added the 1/4s to 1/2s. Great explanation! Makes me remember the fun in math

@Avighna 5 ай бұрын

Both proofs are very cool!!

@FTdhTsjgdkdhH 5 ай бұрын

happy lunar new year

@THICCTHICCTHICC 4 ай бұрын

This is an awesome way to explain it. I already understood it but you made it so clear. Thank you

@bprpcalculusbasics 4 ай бұрын

Thank you!

@Bla_bla_blablatron 4 ай бұрын

y'all stole that from black folks

@denielalain5701 4 ай бұрын

Helo! Can you make it such that this sum, or the harmonic series adds up to a negative number, instead of proving that it is impossible? I mean can you approach it with an alien foresight of this summation must end somewhere? What would you tell? What would be the sum of this series, and what would be your approach in order to find it anyway?

@Mike__B 4 ай бұрын

I just noticed on your shelf that you had boxes and boxes of markers, do you get a good price on those? As a teacher who has had to buy his own markers that looks like an absolute gold mine!

@bprpcalculusbasics 4 ай бұрын

I got them on Amazon, ranging from $10 to $13 a box. Not a bad deal in my opinion. Btw, the “bullet tip” is the must! www.amazon.com/dp/B00006IFIN/ref=cm_sw_r_as_gl_api_gl_i_dl_TPD5KRK4A8PDMT9Y1PRR?linkCode=ml2&tag=blackpenredpe-20

@Mike__B 4 ай бұрын

@@bprpcalculusbasics Thanks for the advice, I'll give the bulllet tip a try, I've exclusively used the chisel tip and yeah you gotta hold it at the right angle to write nicely.

@bprpcalculusbasics 4 ай бұрын

Right. And for the bullet tips you won’t need to worry about that!

@kckcmctcrc 5 ай бұрын

This is a brilliantly simple proof.

@connor5890 Ай бұрын

Thats such a cool proof

@philippenachtergal6077 5 ай бұрын

An intuitive (not rigorous) way to see this goes like this: If you take all elements between index 10 and 100, there are 90 of them and they have a median value of 1/55 so you expect the sum of them should be above 1 If you take all elements between index 100 and 1000, there are 900 of them with a median value of 1/550 so you expect the sum of them to be above 1 And you can go on like that for infinity. The above is not a proof because the median value can be bigger than the average value but it is intuitive and you get the feeling that there has to be a proof that works "kind of like that". And indeed, its not hard to change that into a proof as each of the steps above can easily be proven to have a value > 0.9 by replacing all its terms by the last term. This is quite similar to the usual proof made in the video.

@WhiteGandalfs 4 ай бұрын

Essentially, that's the same as he does but with base 10 instead of base 2. I think this one is better suited for children that are not yet familiar with base 2 thinking.

@AbramSF 4 ай бұрын

He didn’t take the median value in this proof with 1/2 though. Each power of 1/2, he took the minimum value. Which allows any sum of those numbers to be bigger than the minimum.

@renem.5852 4 ай бұрын

Just don't look at median value, but the minimum value of your intervals. If you have 90 * (sth that is at least 1/100), you have at least 90/100 = 0.9. If you have 900 * (sth that is at least 1/1000), you have at least 900/1000 = 0.9. ... Even though I don't see how this is more intuitive than taking the powers of 2, taking an obvious lower limit should be more obvious than the median for most people.

@philippenachtergal6077 4 ай бұрын

@@renem.5852 Of course, and that's what I explain at the end. I was showing a thought process, first I think about adding terms in an exponentially increasing batch size and I use the median value to get an idea of the partial sum because that's what first came to mind and it seems "about" right. This is enough to tell me that there is probably a proof not for away. And you get an actual proof by using the minimal value rather that the median(that doesn't bend well to prove a boundary value) or the average (because you can't easily figure out the average value). When you explore a problem, you don't necessarily get to a proof right away but misses can still be good enough to confirm some ideas and guide you toward a proof.

@iankeith 21 күн бұрын

The second this clicked I laughed, it's a dang cool proof.

@ColemanMulkerin 4 ай бұрын

When I think about the fact that the nth term of the harmonic series is almost equal to the (n + 1) th term while in the power of 2 series the (n + 1) th term is half the nth term it helps it make sense to me.

@magicmulder 4 ай бұрын

The sum over 1/n^s goes to infinity for s1.

@mccask 4 ай бұрын

Taps the board, it magically erased!

@viathorr 3 ай бұрын

I never thought I would be watching this for entertainment at 4am🙂

@exponentmantissa5598 4 ай бұрын

That was a nice simple proof

@Itz_JoshDM 3 ай бұрын

haven't done calc in a while. this was a nice refresher.

@grimtapestry5585 4 ай бұрын

Such a satisfying proof, I'm wondering how this factors into harmonics in the real world which is quantized. Or in other words, is there a limit to how much can a wave be fractionated.

@MrRabix007 4 ай бұрын

When i saw the miniature i thought that video is about Divergence of the sum of the reciprocals of the primes . 1/2+1/3+1/5+1/7...+1/31 but it is complicated to prouf

@barnabusowl4252 5 ай бұрын

Even more peculiar, if you revolve the graph around the x-axis and then evaluate the area formed by the curve from 1 to infinity given by the integral (pi/x^2) dx, you get the finite area of pi 🤔

@MikeGz92 5 ай бұрын

I don't remember this demonstration, but it's great 🤩

@forbidden-cyrillic-handle 5 ай бұрын

Nice name 🤣

@user-yx2en7ei4d 4 ай бұрын

@@forbidden-cyrillic-handle lolll it must be italian

@AdventCloudStrife 4 ай бұрын

Does this constitute a regrouping of terms in a series that isn’t absolutely convergent, and so not a valid thing to do?

@fafflerproductions 4 ай бұрын

I'm gonna be honest, I just watch for how quick you swap marker colors

@roger7341 5 ай бұрын

Nice. Making something littler BIG.

@brettski74 2 ай бұрын

This got me thinking, though. The increase in the sum of this series between successive powers of two seems to converge. I'm wondering what the limit may be for the sum of 1/m from m=2^(n-1)+1 to m=2^n as n approaches infinity. I calculated it up to m=1025..2048 and it seems to be approaching a value somewhere around 0.693, but not sure if this actually converges on a finite value or if it just goes off to infinity even more slowly than the harmonic series.

@nosirrahx 29 күн бұрын

Is there a function that describes how many iterations it takes to get => than each integer? For example, how many iterations does it take to get over 5, over 50, over 500 ......?

@socks2423 5 ай бұрын

So where does it break where the series converge?

@457max 22 күн бұрын

I liked that integral proof.

@o_s-24 3 ай бұрын

I hate this fact. This series has business being divergent, but here we are

@9adam4 3 ай бұрын

Clearly the series should be exiled from the system and mount a resistence. 😊

@SteveThePster 4 ай бұрын

Can you do a video proving why the sum of 1/(x^n) converges whenever n>1 ?

@kelecsenyizoltan274 3 ай бұрын

Nice!

@fluffyfang4213 4 ай бұрын

Upon seeing the thumbnail (but before watching the video), I ended up doing similar logic but in multiples of 10 instead of 2. So like, set all the fractions from 1/2 to 1/10 to equal 1/10 each. Add them up and you get 9/10. Do the same for fractions 1/11 to 1/100, setting them all to 1/100 and you get 90/100... or 9/10. Glad to see I was on the right track even if going by multiples of 2 is way easier to explain!

@Catsy-cy7yi 4 ай бұрын

As a current 8th grader, I am SUPER HAPPY that you included the integral example! I couldn’t quite understand the first explanation but the second one made me understand in a split second! I LOVE that you include more than one way to solve equations! Thank you!

@nguyenbao8987 3 ай бұрын

Not the 8th grader understanding integration... Aus education is doomed

@Catsy-cy7yi 3 ай бұрын

My Turkish father taught me some simple precalculus 3 years ago, I can still remember lots.

@redrevyol Ай бұрын

Now it makes sense how integrals are used in infinite sums or series. I'm too lazy to open up my calculus book to learn how to do alternating sums for cos and sin though.

@Frozen_Hope 2 ай бұрын

The marker swapping feels like a magic trick, that's some skill

@FM1234 4 ай бұрын

Such a cool proof

@adw1z 5 ай бұрын

Fun fact: H(n) := 1 + 1/2 + 1/3 + .. + 1/n. Then, lim n->∞ [H(n) - ln(n)] = γ , where γ = 0.577216…. is the Euler-Mascheroni constant. This shows that the harmonic series H(n) grows asymptotically like a shifted up log function, which is indeed (albeit very slowly!) divergent as n->∞. We can use this to estimate H(n) for large n, via: H(n) ≈ ln(n) + γ + 1/(2n) + O(1/n²) (this is the asymptotic expansion of H)

@methatis3013 5 ай бұрын

Also, an interesting fact, it isn't known whether γ is even rational or irrational

@zachariastsampasidis8880 4 ай бұрын

Wasn't it shown than γ and e^γ are algebraically independent

@mtaur4113 5 ай бұрын

Idea number one generalizes to Cauchy's Test for monotonic series. The series of a monotonic sequence a_n converges if and only of the series of 2^n a_(2^n) converges. The harmonic series is a model case for the test. As a bonus, you could show that the integral of ln diverges, even if you did not know that it was the inverse function of exp, using you method 1, or Cauchy's Test on the the associated series.

@GynxShinx 3 ай бұрын

Very funny to think that the whole front of the harmonic is finite no matter where you cut it off so if you start at 1/1000000000 and then continue the harmonic series +1/1000000001+1/1000000002... and so on, it still diverges to infinity.

@AkshaySinghJamwal 4 ай бұрын

What a beautiful proof.

@clintonweir7609 5 ай бұрын

I was thinking about it. For the sum of the series 1/n for n=1...N, you can create a fraction with a denominator that is N! and the terms are, I think, N!, N!/2!, N!/3! etc. And then I realized that I was just working my way back to the original problem. Whoops!

@alessiodibella2003 3 ай бұрын

THIS FINALLY MAKES SENSE TO ME! With the demonstration with integral calculus, I could not be convinced that 1/2 + 1/3 + 1/4 + ... could diverge, thank you very much!

@kingjulian1202 3 ай бұрын

Why am I watching maths at midnight 😭😭

@MurderMostFowl 4 ай бұрын

This is a special case, for 1/X^1,right? (Well and 1/X^0 too) Any other positive integer power of X will converge? Or do I have that wrong?

@4.0.4 4 ай бұрын

First time the numeric proof is more intuitive (to me) than the geometric proof!

@GrandMarshalGarithos 4 ай бұрын

Yet the series 1/n^2 does converge, I wonder what the turning point looks like. If you had 1/n^x at what x does the series switch ?

@quentind1924 3 ай бұрын

That's the Zeta fonction. And i'm pretty sure it's finite for any x>1

@colinjava8447 4 ай бұрын

Well S = 1/1+1/2+1/3+...+1/n > ln n, and ln n diverges, so so must S. It does require proof that S>ln n though.

@guymorag8380 2 ай бұрын

You can confirm this with the integral test!

@tflcn 4 ай бұрын

Infinite amount of mathematicians walks into a bar. First orders 1 pint of a beer, second orders 1/2, third 1/4 etc... barmen silently pours 2 pints

@TheMasterGreen 2 ай бұрын

another reasoning for the geometric proof is that since 1/x has a horizontal asymptote at 0 it never reaches 0 but it approaches it as x becomes larger and larger. Therfore the "infite" term would be slightly greater than 0 (also known as nth term test) so the final series or sum will not converge because you are always adding to it forever.

@ironeche3702 2 ай бұрын

What about sum( 1/2^x) x→inf

@Billy_Vanner 4 ай бұрын

Once I figured out how it worked I realized this dude has so many markers

@elielmachado 3 ай бұрын

I don't know how to integer, but LOL for the inf - 0.693. I mean is just one small step from paradise, but also in heaven. It's like the car at the light speed with the headlights on

@fawwaznawaz1698 4 ай бұрын

why am I getting motivated by numbers 💀

@yagan_dawn2008 4 ай бұрын

Amazing

@vikramvilla 5 ай бұрын

Awesome. For a series to give a finite sum, it must converge. HP doesn't converge, so the sum is infinity. Your proofs are really cool.

@tyuh860 4 ай бұрын

What is converging? I only started line functions in math right now, so I haven't had this yet.

@vikramvilla 4 ай бұрын

@@tyuh860 converging means if you keep on adding the numbers in a series, it shall get closer and closer to a finite number.

@Regian 3 ай бұрын

Converging is like mathematical edging

@A1readyDead 4 ай бұрын

I have always hated math and yet here I am at 3am watching a match video

@oyuyuy 2 ай бұрын

I think a valuable and quick thought experiment is thinking about the sum between '1/1000... --> 1/3000' for example. It' adds up to ~1.00 which is not an arbitrary amount. Same for 1/10.000... --> 1/30.000

@cose3541 3 ай бұрын

how many number like these in this sequence would it take to reach 10? this is fascinating that is infinite.

@gtothereal 4 ай бұрын

Bruh the marker skills.

@luitmeinen1902 4 ай бұрын

Very interesting! Question about the first proof though: didn't you prove there that it is bigger then infinity? Because you made everything smaller (1/3 -> 1/4 etc). Or is this one of those cases where it is not bigger because the infinities are the same (if you know what I mean)? Different types of infinities tend to confuse me 😅

@ianlindstrom2019 4 ай бұрын

That's the cardinality of an infinity, which is a separate "size" concept yes (NOT the same thing as the size of a number!) Basically, infinity isn't a number, so if you show that something adds up to greater than infinity, you show that it must also be adding up to infinity, but have shown nothing about its cardinality. The information of how big the number is disappears, because it ceases to be a number.