Matt meets Jordan Ellenberg: BONUS FOOTAGE
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- čas přidán 13. 06. 2015
- 1 + 1 - 1 + 1 - 1 + 1 - 1 +... = ?
1 - 1/2 + 1/3 - 1/4 + 1/5... = ?
Matt and Jordan continue to chat in the Penguin offices, covering Grandi's series and the Alternating Harmonic Series.
Watch the first video: • Matt meets Jordan Elle...
Find your local MathsJam: mathsjam.com/
Jordan Ellenberg's book: www.jordanellenberg.com/how-no...
Matt Parker's book: makeanddo4d.com/
Music by Howard Carter
Design by Simon Wright
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
New book: makeanddo4D.com/
Nerdy maths toys: mathsgear.co.uk/ - Zábava
![Matt Explains: The Lottery [featuring: Choose Function, Infinite Geometric Series]](http://i.ytimg.com/vi/lP58mP8Wchc/mqdefault.jpg)
Oh, I got a surprise plug at the end!
+singingbanana +standupmaths couldn't you say that the difference between the original sum of 1+1/2+1/3... is different from the later value by 1/2*(twice it's own value it's own value) = it's own value and have that twice it's own value is then show that it has no value that makes sense because it shows 0 =ln(2) if I did the math correctly ?
That could've been worded better...
+Noel Goetowski surprise plug in the end*
+Vulcapyro That could be misinterpreted...
jaaaaaames!
I like that "Watch part 0.9999..." annotation xD
What you mention there around 10 minutes in, about finding and exploring things that are already known, is pretty much what amateur astronomy is all about. Millions of people buy expensive telescopes and astronomical cameras to both see with their own eyes and capture on camera, objects that have been photographed by professional astronomers in much better quality many times over already. But it's still fun and exciting to do it yourself! :)
the number _e_ is like a stalker... You're just minding your own business, doing some casual math when suddenly... BAM, _e_ or his (anti-)twin brother, _ln(...)_ appears. No matter what you do, no matter where you go, they just keep popping their heads up. I personally think they're even worse than their cousin, _pi_. _Pi_ just isn't as sneaky most of the time.
I think _i_ is a much nicer number. _i_ has the decency of making sure you know when he's showing up.
Niosus Because apparently numbers are male..?
JNCressey Funny. I've always thought of i as male and e as female. No idea why. :)
LOLwut? They're numbers, they don't have genders.
JNCressey English needs a gender-neutral pronoun that still personifies the subject. Having to write he/she all the time for the neutral case is really annoying.
Numbers can be any gender they want, and they can love any number they want. :)
Niosus Singular 'they' is fine. 'He/she' is ridiculous.
I'm really surprised that Matt hadn't heard of the alternating harmonic series.
me too 0.0
+Jose Villegas I think he was just pretending not to know :p
It's also easy to forget
You know nothing Jon Snow
+Taylor Foulkrod I think it really shows the humilty that Matt has, even obviously knowing.
Back when I was in college in 2002-2006, I found something interesting which I know think I was the first to stumble on to. I might still have evidence to back this up, because I posted a question to an online forum (Dr. Math, I think) where I asked if the sequence:
3^2 + 4^2 = 5^2
3^3 + 4^3 + 5^3 = 6^3
...
3^n + 4^n + ... + (n+2)^n = (n+3)^n
Was true beyond the first two (which are easily demonstrable.) The answer I got back was "no, this is just an amazing coincidence." Which I told them in my question, that I had found an amazing coincidence and was just wondering about the general case.
So, out of fun from this video, I checked out the first few "residuals" from that, and got this sequence: {0, 0, 143, 3793, 84542, 1919704, ...} and decided to check it out:
oeis.org/search?q=0%2C0%2C143%2C3793%2C84542%2C1919704&language=english&go=Search
Discovered in 2012. I feel so broken hearted. I was actually onto something new, and I just walked away from it because they told me "just a coincidence."
+nychold i think you should look at it with a different view, you basically found something yourself, which was proven to be atleast something, so this could happen again, you could stumble over something great and maybe then you even will be recognized :)
If being recognized was the sole incentive that drove your maths..
@nychold I do not believe you were onto something, the comment on the page you linked even says "The first two terms of the series illustrate the famous equalities 3^2 + 4^2 = 5^2 and 3^3 + 4^3 + 5^3 = 6^3. The following terms show how this eventually diverges."
What it's actually saying is that
(3^n + 4^n + ... + (n+2)^n) + a(n) = (n+3)^n
where a(n) is some unknown function that can be approximated with a(n) ~ k*n^n, where k = e^3/(e-1). It really was just a coincidence that your series follows (n+3)^n for the first few terms without the additional function a(n).
Those annotations ... "Watch part 0.99999 ..."
honestly would love to see them make another video together, this video is a classic
love you matt for the "latural nog". i'm slightly dislexic and i say that all the time
The thumbnail is different: 1+1-1+1-1+... Which alternates between 1 and 1.5
We need more of these, please!
Learning Calculus is what truly made me fall in love with e and natural log. How can we model the rate of change of e^x? Well, with e^x. And the rate of change of that? Quite obviously e^x. It's a number so synonymous with rate of growth that the rate of growth of the rate of growth is the rate of growth itself. Natural Log is just as beautiful. What is the model for the rate of change of ln(x)? My god that must be some horrible complicated equation. Or it's 1/x.
In the ancient days, people may have struggled modeling rate of change, but now we can do it with e's.
I just love these videos. Maths jams sound awesome! Too bad I haven't really found one in Nebraska.
So Classical! Great Job!
Awesome, I recognized that last sequence as ln(2)! I'm pretty sure we used that sequence at some point in my education, because I'm definitely not a human computer.
ln(2)! ≈ 0.90735801689
Hoo Dini he's not saying ln(2) factorial, he's saying ln(2) with an exclamation mark
1-1/2+1/3-..... Is just the maclaurin series's for ln(1+x) where x=1 i.e ln2
I just enjoyed every second of this video.
I stopped as soon as you said you wrote a program to calculate that ln(2) series and wrote my own for the heck of it. Just finished crunching 10^10 iterations, accurate to 11 decimal places!
Awesome video. Keep up with the good work ;)
Great pair of videos!
We know infinite rationals can sum to an irrational (e.g. via series expansion of e^x) so maybe we shouldn't be too surprised if something like 1+1-1+1-1+... can land us with a fraction, or 1+2+3+... (weep!) can send us into negative space!
Of course, the series expansion of e^x is also what's behind another alarming maths "number type mismatch": sending us onto the Real number line when calculating cosines of imaginary numbers!
Sold! That did it for me.. I'm a subscriber!
I love the "Watch part 0.9999... here" link in the beginning. Subtle but hilarious
You can make sense of the Grandi series taking on the value of 1/2 in several ways. One way is by talking about equilibrium point convergence. If you consider the sequence a(n + 1) = 1 - a(n) for all natural n and some arbitrary real a(0), then we can talk about the convergence rate of a(n) as a function of a(0). As it happens, a(n) = a(0) if 2 | n, and a(n) = 1 - a(0) otherwise. Hence, a(n) diverges with a two-cycle for all values of a(0) except a(0) = 1/2, in which case it converges to 1/2. Hence a(0) = 1/2 is an unstable equilibrium point of convergence. This is relevant, because it explains the strange phenomenon of why different grouping of parentheses in the Grandi series yields different results. The even pairing is equivalent to letting a(0) = 0 and truncating the series at an even value of n. The odd pairing is equivalent to letting a(0) = 1 and truncating at an odd value of n.
amazing video
Yeah, I recognized the alternating harmonic = ln2 too, it's pretty easy to show if you recognize any Taylor series. ( ln(x-1)= 1-x/2+x^2/3-...
Even more awesome!
10:04 i always find it fun trying to program algorithms that i know already exist for the experience
I have seen that in my first class of calculus in undergraduate math.
There is a little more of magic result about that (The demonstration is not included). The first time I saw this I was amaze.
If {a_n} is sequence such as the series s_n = \sum_{i=1}^n a_i, is convergent but is not absolutely convergent series (the sum of the absolute values is not convergent). Then, for each x, real number, it exist a reorder of {a_n} such as the corresponding series converges to x
if you use the sum to infinite formula, S= a/(1-r) where r is the common ratio and a is the first term, S=(1)/(1-(-1)) thus S=1/2
finally 2 books I want to buy :D
"I solved the Riemann Hypothesis but I need your bank information to send you the solution!"
lmao
Richard Feynman discusses doing math in his teens thinking he found a new form only to realise it was already done. it is great that people are still searching through these sequences..
would have been nice to see some more discussion on the 1+2+3+4+5.....=-1/12 stuff like ive seen the numberphile video but it kinda left me wanting more and i couple of videos about the strange results of infinte series had the potential to cover it
Look up the Riemann zeta function
Look up the mathologer video on the sum. He debunks and demystifies what numberphile did wrong in that video. After watching that video feel free to go back to the numberphile video, but take what they say in that video with a lot of salt.
LATURAL NOG
Nice thing I figured out in primary school is that if you know the square of x and want to know the square of x +1, you take the square of x, add 2x and add 1. Of course its a (x+1)^2 = x^2 +2x +1, but I'm quite happy to have figured this out without knowing any algebra.
Hey Matt, are the Maths Jams in London on specific days in the month like the ones in the US? I really like the sound of them and would love to come along...
Haven't you taken Calc 2?
The alternating harmonic series was a prominent part of the series lectures.
did you show that its value is ln (2) by the taylor polomial of ln?
He is obviously pretending not to know, silly.
I like how you at the end refered to "Jordans paper"
Careful with the alternating harmonic series there, it only sums to ln(2) if you add the terms in a specific order! Mathologer has a nice video about it (titled 'Riemann's Paradox'), the gist of it is that this series is weakly convergent, so if you're tricksy with how you add the terms you can make it sum to any number you want. Turns out addition isn't always commutative, whoda thunk?
Thumbs up for Python !
I would love to know about the natual numbers of other number base systems and see them compared and contrasted :D
i got the hard cover of the book how not to be wrong Good book.
you can sum a diverging geometric series using algebra
G = b^0+b^1+b^2+++++
G = 1 + b^1+b^2+++++
G = 1 + b(b^0+b^1+b^2+++++)
G = 1 + bG
G - bG = 1
G(1-b) = 1
G = 1/(1-b)
to test this in the converging case try b = 1/2 or 1/10 as they have otherwise determinable valus
Ahhhhhh- the Parker Identity: G = G. The man is incredible!
The second series is the alternating harmonic sequence, which I'm pretty sure would be covered in a standard HL IB Maths course in high school.
Heather L Kelly I'm pretty sure he would have known to look it up under alternating harmonic series if he had decided to look it up in the first place. Also, description.
Penny Lane True, but they talked about it like it was a surprising result, not something taught in a standard (albeit high level) high school class.
Heather L Kelly Yeah, it was presented in a funny way but when I checked the description that reassured me that he actually knew what he would have had to look for :)
8:34 made me smile
The true test is if there was a physical application of these questions. Without any way to verify it in practice, the arguments only stand up on paper
I think I just proved that the alternating fraction sequence sums to zero. That'll teach me to play.
What do you get if you take the alternating fraction sequence from the standard fraction sequence?
Is this what happens when you take convergent from divergent sequences?
Since you were talking about infinite series you could have talked about the sum of all integers to be a negative fraction, but otherwise great and fun video.
That 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... series has another cool property : you can rearrange it so that it converges to any other real number you like. Any. Real. Number. You. Like. Just by changing the order of the terms.
It's the Taylor series for ln(1-x). Just plug x=-1 and you get the ln(2).
Once In college I "discovered" l'hopital by myself wjile studying to a test. In the question where we had to solve limits the professor said that answers using l'hopital would be considered wrong(he did that for students repeating the subject woudn't do the easier way, he wanted the full process). But I didn't know what l'hopital was because I had never studied it, so iIproceeded to do it my way and ended up getting a zero in the question.
Aren't these on any Calculus 1 text book already? Chapter Series and Sequences.
Is simpler: because they are infinite operations there are the same number of sums than of subtractions they cancel out so that infinite operation os fue first number
The way I like to think of this is a lamp, 1 = on 0 = off. If you switch it on and off an infinite number of times is it off or on?
The first time Matt met Jordan, AND the first time Jordan met Matt. At the same time. Wow ...
+Egil Sandnes Small world.
Adding the parentheses changes the first problem if you consider it as a physical manifestation. The original problem states that you start with one of something and then take that away and then add it back. The parentheses that Matt added to the original problem would physically be represented by starting with nothing and adding nothing to it. The parentheses as Jordan added them states that you start with one of something then take away nothing.
These problems clearly work out interestingly when we just use algebra and numbers, but if we think of them as physical events occurring in time and space they don't play by the rules so easily, primarily because of time. The passage of time for a physically occurring event that is sequential, as this problem is, dictates that one event occurs before the other.
With the thumbnail being a shadow puppet of a rabbit, I thought this was going to be about the math of shadow puppets. I was slightly disappointed.
I can't help but feel this is related to the Riemann zeta function somehow
So is this guy the love child of the creators of south park or something?
They could add that the result of 1/2 seems the most "logic" because what ever the groupement you do for G at the left of the equation it will be the same in the right of the equation; that means G will be "realy" 1/2
I am so uneducated in Maths, and don't have the time to read all comments. My issue is with the manipulation of the sequence itself. So 1-1+1-1 to infinity will lead you to an inconclusive number, as how do we know where infinity ends? But.... when we add in these parenthesis, we are changing the true sequence, we are adding in an extra step, extra junction to judge the behavior of these numbers. Please tell me why I am wrong.
What was the website to check a number? Couldn't hear him.
Anyone have a link to it?
Order of operations dictates you calculate "as is" which makes it unsolvable because infinite calculation is required. The answer to 9 significant digits is therefore "Hmmmmmmmm" :D
3:54 ... You've broken my brain
Anyone else keep hearing "Coach Z"?
Why does the question of 1+1-1+1-1... never seemed to be approached with the laws of order? (Decision before multiplication before addition before subtraction)
And why, if you do this, is ...-1+1-1+1.... focused as ...(-1+1)+(-1+1)+.... instead of ...-(1+1)-(1+1)-(1+1)-.....
The values converge on, or maybe we should say float between, three values 0, 0.5 and 1. Why can the question not have three right answers? Square roots have two answers...
My first guess when I saw that series was 1/e.
I think we can not treat infinity as a number (stop point) it is a concept
You hadn't seen the ln2 infinite series?
Lol, "Normal people." 11:37
Matt, have you seen this? How to use Ln(2) to get π ?
So... if 1-1+1-1+... = G
Then G = 1-(1-G)
Which is G = G
So then G can be any reel number?
So maths is like light, but bet better! Because light is two things at once (a particle and a wave), and maths can be three things at once. Could there be an infinite serie with more than four answers, or is there a limit of three (two numbers and their average)?
11:12 Ah, noticed the extra R that Brits put in. Watched a Lindybeige vid recently, and when Brits say a word that ends with an A, and the next word starts with a vowel, they add an R. "The idear is"
No cesaro sum?
1-1+1-1+1.... I believe can also be written as 1+x+x^2+x^3+x^4....., where x=-1 -> 1/(1-x) = 1/2
Yes using Cesàro summation you can use that formula even if |x|≮1
I haven't come across Cesàro summation before, time to do some reading : )
I just came to comment about the bunny rabbit in the thumbnail.
If you do order of operations it would be 1-2-2-2-2...
What happens when you flip signs?
-1 + 1/2 - 1/3 + 1/4 - 1/5... = ?
Its 4:00 in the morning here and I'm to tired to go at it myself.
***** Then of course you get -ln(2).
the reason grandys series doesn't work is because you're changing the repeated term,
first equation: G=1-1+1-1...
G=(1-1)+ (1-1)...
second equation: A= 1-(1-1)-(1-1)...
A=1-G
therefor G=0 & A=1
2/3+4/9-8/16...?
oh darn it's just 3/5
G=1-1+1-1+1-1... now add G
2G=(1-1+1-1+1-1...)+
(1-1+1-1+1...)
They cancel out so again 2G=1. Two points for G=1/2
The teorem is
en.wikipedia.org/wiki/Riemann_series_theorem
I discovered the Ulam-spiral on my own.
And boy was I disappointed when i realized it already was a known prime-property.
I lack skills. What happens if you flip the signs to 1+1/2-1/3+1/4...? I know it isn't new, just curious.
The 4th answer is the answer exists in a super-position of 1 and 0.
The reason it doesn't have a set answer is simply because it's ambiguous. it depends where you put parentheses.
the series itself has no meaning. its like saying "what is 2 plus ×?"
Laurelindo Where's your Abel prize? O wait...you haven't proven or disproven anything about this series. Just trying to sound smart and failing.
schrodinger's series
Shadow of the Jackalope in the thumbnail.
How come that the first meeting relation is symmetric? Revelation !!
What's the problem? Everything makes perfect sense if we say: the infinite sum of zeros equals 1/2. Now you can put the brackets how you like and get always the same result of 1/2
1+2-1/2+1-1/4+1/2-1/8+1/4
1-1, -1+1, -1/2+1/2, -1/4+1/4
but there's still a 2 and a 1.
answer is 3 according to python..
1-1/2+1/4-1/8...?
oh darn it's just 2/3
THis is what i think
(1-1)+(1-1)+(1-1)...........=0 when there are even no. of 1
in case no. of ones are odd , you can't pair then up in groups ie u have to leave one 1
which makes it like (1-1)+(1-1)+(1-1).........+1=1
Anybody got my point ? :)
Matt, do you happen to have any love for Ruby?
👍👍
Surely you have to state what the sequence ends on, is it +1 or -1? writing ... only causes issues
you must add first? If you bracket them, you are not adding first.