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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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...
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0:00 Why does the harmonic series diverge?
1:41 The classic proof
6:00 Integral test proof
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#calculus #bprpcalculus #apcalculus #tutorial #math
"If you add infinitely many positive number you get positive infinity"? In fact, it’s not! kzhead.info/sun/drOeiaexaJahgKc/bejne.html
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!
...and sheeeeet
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.
It's just slow, but it goes "to infinity" through sheer fraction will.
the sheer fraction will when I introduce 1/n^2
The indomitable fraction spirit
@@hardboiledegg2681 well doesnt this just converge to pi^2/6 ?
@@Saber1320 yeah that's the point, this one would converge despite "sheer fraction will"
@@hach1kokothe futile fractional will vs the unstoppable power of a square
Answer: It approaches infinity incredibly slowly.
*infinitely slowly
Logarithm: "hold my beer" ... "It's going to take awhile"
... can things approach infinity any slower or faster than something else?
@@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.
Marker switches so smooth
yea hes very good at it
Thanks!
I ignored the math completely and was just mesmerized by his sleight of hand color changes.
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.
@@diggoran ***mind blown***
I haven't done math proofs or calc. since college but KZhead has correctly started promoting this content to me and it's really nice to see these clean demonstrations.
Same I’m an accountant I got into school with this 😅
math is Racist! Y'all betta go back to da Crackerbarrel
probably means you share interests with college calc students!
Can you teach a proof why pi is irrational, like assuming it to be rational and prove it with contradiction
Mathologer has done a really good video on that. It's quite a bit more challenging than this problem.
kzhead.info/sun/l6qEkrqMf31qp40/bejne.htmlsi=n08QZRrYRXzSRdfM
That exercise is trivial and left as an exercise for the reader - Fermat probably
kzhead.info/sun/l6qEkrqMf31qp40/bejne.htmlsi=01bxPVz8EVOpP7E-
@@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
This proof is due to Nicole Oresme, a medieval mathematician, theologian, and philosopher.
mfs really just had “smart” as their job and worked in every educated field
@@nchappy16 So true. And the fields weren't as distinct from each other then.
@@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.
Would be good if he proved God
@@bardsamok9221Saint Thomas Aquinas already did that tho
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?!”
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.
chocolate pudding
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)
Lol, that second one seems counterintuitive but actually makes sense thinking about it.
@@matc241why would the second make sense?
I don't understand
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.
@@theeraphatsunthornwit6266 i agree yeah, it also doesnt make sense considering there are more multiples of 9 than there are primes...
The first proof that you showed is really beautiful! Thank you!
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!
This comment is misleading, I still don't enjoy vegetables after watching the video.
she toddlers on my vegetable til i math
Goo goo gaa gaa daada baabaa
Bruh how can one not enjoy vegetables, that stuff is addictive af@@TheAlienPoison
@@Matt-lv1jl dog what veggie you consuming
Even getting to a million would take an incredibly large number.
Because the amount for each next ½ doubles it would take 2^1999999 for the *final ½ alone*
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
@@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;
'e' is omnipresent
@@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
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.
Haven’t seen the second integral proof before - really like it! Great explanation
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.
π and e are omnipresent man
@@vanshamb There's circles involved. Not that weird that pi appears.
@@galoomba5559 ohkk then just e is omnipresent ig
Infinite surface area, but finite volume. You could fill it with paint, but can't paint its sides.
@@CliffSedge-nu5fv daym that makes no sense, maths wrong
Keep up the good content man. You are very regular, and I like that!
Ayo the pfp?
@@customlol7890 coincidence
you #1 Racist
I saw this demonstration in my Real Analysis class 10 years ago. Very elegant.
I saw that proof in middle school.
@@cbunix23 In middle school I didn't now what x meant LOL
@@cbunix23You must live in some advanced alien civilisation if you’re doing these proofs in middle school…
@@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.
@@cbunix23Sheet, what kind of alien civilization taught spherical trigs in high school even before ww2?
Sweet video dude, thanks for sharing. That was cool and informative
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
Y'all stole that from Africa! Or maybe you never heard of Wakanda.
@@Bla_bla_blablatron we wuz kangs
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.
haven't done calc in a while. this was a nice refresher.
One of the coolest proofs I’ve seen, love it - thanks!
Thank you. I had not seen this proof represented graphically before. Now it makes sense.
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.
Informative! Also I love your voice so much
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!
This is basically a proof via a direct comparison test. If b_n > a_n and b_n diverges, a_n must diverge. Never thought of it so much, i just accepted it via the p-series test, where p=1 (the exponent on 1/n) means it is divergent. Maybe you could do a proof where p
The sum over 1/n^s goes to infinity for s1.
Shout out the limit comparison test and integral test (in order according to powering). This video is literally amazing
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.
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.
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.
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.
@@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.
You somehow made it sound simple *and* made me feel smarter just watching 😊
This was a great explanation!
Another amazing video as always
Love it man. Great proof.
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)
Also, an interesting fact, it isn't known whether γ is even rational or irrational
Wasn't it shown than γ and e^γ are algebraically independent
This is an awesome way to explain it. I already understood it but you made it so clear. Thank you
Thank you!
y'all stole that from black folks
You, Sir, have my upvote
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 🤔
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!
Learned this in calc class yesterday. I think my phone is listening into my classes.
so fresh and so clean🎉
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
Great video, very well explained
I didn't noticed his second marker at first and got confused on how the color suddenly changed without putting down his arm 😂
You didn't know what the channel name, bprp, stood for, did you?
@@digitig nope just saw this video in my recommendation and decide to click it on a whim
Man, this is why I watch these even though Im not a math lover. That last bit was the first time someone showed me the point of an integral.
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.
Wow. This was a great explanation.
That geometry proof is nice
Awesome video; very clever proof
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
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
Both proofs are very cool!!
I hate this fact. This series has business being divergent, but here we are
Clearly the series should be exiled from the system and mount a resistence. 😊
This is a brilliantly simple proof.
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.
What about sum( 1/2^x) x→inf
In having a hard time understanding how the inequality holds after we are done with the derivation. I understand the logical steps but at the end we end up with inf > inf. Is this correct? I know this doesnt have to do anything with the actual size of infinities as they have the same cardinality. Plugging inf>inf into wolfram alpha returns false but idk if its correct. How does the inequality still hold? Thanks for the video!
We end up with something greater than inf, then that something can only be inf.
Wow great explanation ty
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.
1 > 1/2 1/2 + 1/3 > 1/2 1/4 + 1/5 + 1/6 > 1/2 1/7 + 1/8 + 1/9 + 1/10 + 1/11 + 1/12 > 1/2 Etc. Always the last term is the smallest, and their sum is greater than the number of terms multiplied by the last term.
I don't remember this demonstration, but it's great 🤩
Nice name 🤣
@@forbidden-cyrillic-handle lolll it must be italian
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?
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!
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
@@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.
Right. And for the bullet tips you won’t need to worry about that!
Nice. Making something littler BIG.
happy lunar new year
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
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!
Not the 8th grader understanding integration... Aus education is doomed
My Turkish father taught me some simple precalculus 3 years ago, I can still remember lots.
You can confirm this with the integral test!
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!
I'm gonna be honest, I just watch for how quick you swap marker colors
It is also true that if you peel out just the odd denominators : 1/3 + 1/5 + 1/7 + .... This series also diverges to infinity! Same for the even denominators: 1/2 + 1/4 + 1/6 + .... diverges to infinity Even more bizarre, just take the odd prime denominators: 1/3 + 1/5 + 1/7 + 1/11 + 1/13+ 1/17 + 1/19 + ..... also diverges to infinity! Just think about how slowly this must diverge.The concept of infinity will stretch your mind.
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.
The marker swapping feels like a magic trick, that's some skill
If you put prime numbers on the denominator, the term approaches 0 even more quickly But the series still goes to infinity.
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.
What is converging? I only started line functions in math right now, so I haven't had this yet.
@@tyuh860 converging means if you keep on adding the numbers in a series, it shall get closer and closer to a finite number.
Converging is like mathematical edging
That was a nice simple proof
1+2+3 converges at -1/12 but 1/2+1/3+1/4 diverges, something is clearly wrong here.
The sum of the integers doesn’t converge under normal circumstances. Under a different definition of convergence or of summation then sure, but there’s a good chance that the harmonic series converges in that system too.
@@ConManAU One definition/system used for these types of convergence is the analytic continuation of Riemann's zeta function (zeta(s)) to the complex plane. In this definition, we have indeed zeta(-1)=-1/12. Funnily though, 1 is a pole of zeta(s), and zeta(1) happens to be the harmonic series. So also under this definition the harmonic series is not defined!
Saying it's equal to -1/12 is one thing, but saying it converges is the most illegal thing I've heard.
A guy on mathologer channel show -1/12 thing is wrong, basically
@@theeraphatsunthornwit6266 it's not wrong. What's wrong is saying that the sum converges to that value. It can ve regularized to that value tho.
Integral from 1 to infinity of 1/x. This works because 1/x is continuous, decreasing, and positive. When you do that you get ln(infinity) - 1. This equals infinity - 1. That is infinity!
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?
Can you do a video proving why the sum of 1/(x^n) converges whenever n>1 ?
1/2+1/3=5/6 5/6+1/4=13/12 In just three fractions you increase above one. I'd just as readily assume that the number converges to any positive value as infinity based on that information, but it's cool to know the answer in more detail
Another short way is Σ 1/n > Σ_{p prime} Σ_i 1/p^i = Σ_{p prime} 1 , which is infinity because there are infinite primes
this dude's got me watching a math video for entertainment
Why are you allowed to subtract ⅓ from one side, and then say it's less than the other side? Isn't ♾️-⅓ still ♾️? How can ♾️ be less than ♾️?
1/3 isn't being subtracted... it's being replaced with 1/4 as 1/3 > 1/4. It simplies the problem and make a new series which is smaller than the origional series. If you can confirm that the smaller series does not converge, then the bigger one must not converge either. No one is subtracting from infinity, infinite is not a mathematical operator or value you can do math to.
Harmonic Series :)
Taps the board, it magically erased!
How does it go to infinity if it requires component numbers larger than is sum to produce it past 1. 1=1 1+1/2=1.5 1.5
someone needs to rewatch the video
Great video demonstrating the proof but we should make sure to make the distinction that infinity is not a number. Infinity minus a real number is meaningless.
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.
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.
Such a cool proof
Once I figured out how it worked I realized this dude has so many markers
What about this sequence (1/1)+(1/10)+(1/100)..and so on? it should result infinite because you are adding numbers to infinite but at the same time you are stuck with adding 1 decimal every time 1/1 = 1 1/1+1/10 = 1,1 1/1+1/10+1/100 = 1,11 1/1+1/10+1/100+1/1000 = 1,111 1/1+1/10+1/100+1/1000+1/10000 = 1,1111 What am i missing?
It would go to positive infinity. +1/5 + 1/6. Later on +1/1,000,000, +1/1,000,000,000... Though the numbers added are decreasing in size they are being added without a limit and therefore the answer to the sequence is Positive Infinity.
I have always had the same question: why cant u use the same reasoning for 1/x^2, for example? in the end, u have infinite terms too, so u could group them the same way. U will just need more terms I know this is wrong cuz pi^2/6, but idk u cant do the same. Could u explain pls?
It’s a good question, and to explain why this proof doesn’t work for the reciprocals of squares notice that if you try and do the same trick you can’t actually find a way to group the terms together to give an infinite number of things that are all bigger than 1 - if you look for all the terms that are bigger than 1/M, there will be fewer than M of them. On the other hand, you *can* do something similar to the area proof to show that the sum of 1/n^2 is *less* than an appropriate integral of 1/x^2 that has a finite value, which shows that the sum itself converges.
1+1/2+1/4+1/9+1/16+1/25... Notice how this is larger than 1+1/2+1/4+1/16+1/16+1/32+1/64+1/64+1/64+... You don't have enough terms in front of each power of 2 in order for them to cancel out. We had 2 times 1/4, 4 times 1/8 etc in the original problem. Here, we don't have that luxury
interesting. But the most amazing thing is the presenter's mental agility in swapping pens seamlessly and flawlessly!
I have a much more simple demonstration (beware: joke ahead) -> if you have 1/∞ and sum it up an infinite amount of times, you roughly have ∞/∞ so around 1. If you keep up with the sum, you can basically sum 1 an infinite amount of times, which..... Bring the sum to infinity!! 😂
why am I getting motivated by numbers 💀
I didn't know that 1/n-1 is as close as 0.5 from infinity; knowing that that the set size is infinity/2. amazing
I think explaining simply the strategy of the proof beforehand helps the student be subconsciously cooperative and allows him to tap into his intuition as he follows the explanation. For me, telling beforehand that the proof is gonna be to show that our sum is bigger than something else that will obviously be infinite. Immediately, the students mind goes in that direction. His attention is focused on the information provided at each line by the firmal proof.
Where can I get that t shirt
My Amazon merch store. Link in description 😃