It might sound strange at first, that you're unable to subtract or add sums and retain equality, but we're dealing with infinite sums here.
To apply basic mathematical operations, you first have to prove that they work as expected.
If they don't, then you might screw up mathematics if you use them anyways.
Assigning values to divergent series is a wide topic in mathematics, as described for example here. "In specialized mathematical contexts, values can be objectively assigned to certain series whose sequence of partial sums diverges, this is to make meaning of the divergence of the series."
As said, whether you accept those methods and the conventions that they use as "valid" is up to your mathematical-philosophical preferences, I suppose.
As said, whether you accept those methods and the conventions that they use as "valid" is up to your mathematical-philosophical preferences, I suppose.
I think the major issue is that if you accept that performing math on divergent series is valid, then you can eventually construct a contradiction. We try to avoid contradictions, because from a contradiction anything may be proven.
An example:
Consider a positive integer n
Construct the divergent series 1+2+3+4+...
The divergent series has a value strictly greater than our integer n, let's call this value k.
However, if the manipulations described are valid, then it also has a value of -1/12 which is strictly smaller than n.
Therefore, we have k > n and k < n. And if k < n then k <= n. So have k > n as our proposition p. We have p and ~p.
Consider some proposition q. p or q is true. Because p is true. However ~p is also true. ~~p or q is true. But ~~p is false. Therefore q is true. And Santa Claus exists.
Build a man a fire, warm him for a day,
Set a man on fire, warm him for the rest of his life.
Consider a positive integer n
Construct the divergent series 1+2+3+4+...
The divergent series has a value strictly greater than our integer n, let's call this value k.
That's incorrect logic. You are just assuming that the infinite sum has a value that's greater than n. On what do you base that assumption?
I understand that you are assuming this because you are thinking about partial sums and limits. Sure, at some point the partial sum of that series is larger than n, and the limit as the number of terms approaches infinity, is infinity. But the infinite sum is not a partial sum, nor a limit statement. You need to forget about partial sums and limits when dealing with them.
I suppose that this is, once again, contingent on which philosophy of thought you want to use. You may want to always think about infinite sums in terms of its partial sums and the limit of these partial sums. But you may also want to consider a sum with an infinite number of terms a completely different beast altogether. Something that cannot be described with partial sums and limits.
The methods that are used to assign values to divergent series seem to be consistent, not arbitrary. It's the reason why eg. S=1+2+4+8+16+... is assigned the value -1 and not something else. (-1 makes logical sense because it is the result you get when you consider S-2S.)
It may be unintuitive and hard to accept, but is it wrong? It depends on your philosophy of thought.
I don't quite understand why you keep talking about some philosophy stuff.
Warp wrote:
It may be unintuitive and hard to accept, but is it wrong? It depends on your philosophy of thought.
No it actually depends on what your predefined ruleset is and which definitions and axioms you follow. Basically, it depends on the mathematical context.
Warning: Might glitch to creditsI will finish this ACE soon as possible
(or will I?)
I think it's better to understand this summation trickery from a viewpoint of a relation. Given a divergent series and summability method (like Zeta function regularization) that relation is the set of all tuples (sum, number) for which the method works (the relation should be right-unique for the method to be well defined), so Zeta-function-regularization(1+2+4+...) = -1.
It's unfortunate that often "=" is used as this already has a rigorous definition in this case (e.g. the e-N-definition of convergence) and by letting <infinite_sum> = S you already imply such a (usually real) S exists, which is good for "back of the envelope" calculations at best. No philosophy involved here.
All syllogisms have three parts, therefore this is not a syllogism.
I don't quite understand why you keep talking about some philosophy stuff.
"Philosophy" doesn't just mean some bearded old guy sitting on an armchair pondering about concepts like the human condition and what is beauty.
One of the dictionary definitions of philosophy is: "the critical study of the basic principles and concepts of a particular branch of knowledge, especially with a view to improving or reconstituting them"
Philosophy deals with things like what is the proper scientific methodology that minimizes biases and errors, and tries to maximize the accuracy and validity of results. It deals with things like logic, logical fallacies, and biases.
Many concepts in mathematics have required a shift in thinking, in the philosophy of mathematics. For example, for a long time mathematicians at large discarded the concept that the square root of a negative value had any kind of significance or use in any calculations; they just considered any calculation that involved it invalid. Of course this changed with newer developments, and now dealing with them is considered pretty much universally non-controversial.
The same can be said of infinities, and their different "magnitudes". Heck, even the concept that the set of real numbers is "larger" than the set of natural numbers was controversial at one point. (If I'm not mistaken, a few professional mathematicians to this day are not comfortable with the thought, and have objections to it.)
Mathematicians seem very divided on the question of whether you can say that eg. "1+2+4+8+16+... = -1". Some are ok with it, others are categorically against it.
How would you call that other than a different philosophy of mathematics?
No I wasn't asking for a definiton of what philosophy means, thank you anyways.
What I actually meant was why you keep bringing up your philosophy stuff when there are already several people that clearly tried to explain how philosophy makes no sense here.
It's like a child who keeps talking about how Santa Claus clearly exists even though it already heard several people trying to prove that fact wrong, but simply ignoring them.
Warning: Might glitch to creditsI will finish this ACE soon as possible
(or will I?)
Mathematicians seem very divided on the question of whether you can say that eg. "1+2+4+8+16+... = -1". Some are ok with it, others are categorically against it.
sum(2x) is infinite when you're using a Totally Regular Summation Method. It only generates a cosmic underflow error when we start using other methods. I'm not good enough with infinite sums to explain how Euler's summation using the general formula of 1 + y2 + y3 + y4 ... = 1/(1 - y) leads to -1 tho.
E: Had it explained to me.
Say S is the sum of 2x
S = 1 + 2 + 4 + 8 ...
This means 2S, double the sum of 2x, would be this:
2S = 2 + 4 + 8 + 16 ...
What happens when we take S - 2S?
S - 2S = 1 + (2 - 2) + (4 - 4) + (8 - 8) ...
S - 2S = 1
-S = 1
S = -1
What I actually meant was why you keep bringing up your philosophy stuff when there are already several people that clearly tried to explain how philosophy makes no sense here.
So I ask once again: Some professional mathematicians are ok with the concept (of divergent infinite sums having a finite value) while others are categorically against it. What would you call that other than different philosophies of mathematics?
So I ask once again: Some professional mathematicians are ok with the concept (of divergent infinite sums having a finite value) while others are categorically against it. What would you call that other than different philosophies of mathematics?
Can you back up this claim that there are [living, or at least, modern] professional mathematicians who disagree with this concept before blazing forward and trying to start a debate about how math (specifically math and not numbers) works?
I am still the wizard that did it.
"On my business card, I am a corporate president. In my mind, I am a game developer. But in my heart, I am a gamer." -- Satoru Iwata
<scrimpy> at least I now know where every map, energy and save room in this game is
Mario Party 4, Hide and go boom.
3 players hide in their choice of 4 cannons, with multiple people allowed in the same one. The other player then has 3 goes to try to find them all, otherwise he loses. What's the probability of winning? What if N players are hiding?
Each enemy player has a choice of 4 cannons to go in (say, A, B, C or D) and at the end the last player chooses 3 of them (say A, B and C), meaning he only wins if nobody entered the one cannon he didn't pick (D).
Since there are 4 cannons, the probability of an enemy player not choosing the leftover cannon is 3/4.
So with 3 players the chance of winning is (3/4)^3 ≈ 42,19%.
With N players the chance of winning is (3/4)^N.
Assuming the enemy players comunicate and always choose the best strategy, the chance of winning goes:
0 enemies - 100%
1 enemy - 75%
2 enemies - 50%
3 enemies - 25%
≥4 enemies - 0%
Warning: Might glitch to creditsI will finish this ACE soon as possible
(or will I?)
That's assuming they communicate, keep it as totally random. When that happens, they'll be overlaps, and thus potentially more than 1 empty cannon. That gives an additional chance for the player to win, if he chooses not to check that one either, and thus complicates the formulas.
With players in ABC, only firing ABC would work. With enemies hiding in AAC, checking either ABC or ACD would be a winning combo.
Hence my interest in larger numbers. Granted if 20 people were hiding inside, it'd be very unlikely that a given cannon remains empty, but still possible to have up to 3 totally vacant and thus some chance of winning remains.
Edit: never mind, wasn't understanding it properly. Far easier than I thought.
When did this thread become, "Do Warp's Math Homework?"
Or, more generally, "Do Warp's Googling?" Nested Radicals, Ramanujan Style.
I am still the wizard that did it.
"On my business card, I am a corporate president. In my mind, I am a game developer. But in my heart, I am a gamer." -- Satoru Iwata
<scrimpy> at least I now know where every map, energy and save room in this game is
http://www.youtube.com/Noxxa
<dwangoAC> This is a TAS (...). Not suitable for all audiences. May cause undesirable side-effects. May contain emulator abuse. Emulator may be abusive. This product contains glitches known to the state of California to cause egg defects.
<Masterjun> I'm just a guy arranging bits in a sequence which could potentially amuse other people looking at these bits
<adelikat> In Oregon Trail, I sacrificed my own family to save time. In Star trek, I killed helpless comrades in escape pods to save time. Here, I kill my allies to save time. I think I need help.
I thought that this was a "math challenges" thread, not a "help me with this problem that nobody has ever solved before and can't be found on google" thread.
I understand "challenge" to mean "can you solve this mathematical brain teaser without looking it up?"
If this is not that kind of thread, then perhaps it should be renamed, to avoid confusion and people throwing sarcastic remarks to those who misunderstand the nature of the thread?
Here is one based on a classic puzzle concept: lasers and mirrors
There is a 2d grid of the size NxM. You are bouncing one laser across all of the tile in the grid. You may place one mirror on each tile. Mirrors are one sided and have to be angled on a 45 degree angle compared to the grid.
The laser enters and exits the grid from outside the grid, each end aligned with one of the grid axis.
How many possible solutions are there?
Joined: 4/20/2005
Posts: 2161
Location: Norrköping, Sweden
Here's a problem I thought up some time ago:
Consider the function p(n) = (-1).^(n-1) * "the n:th decimal of pi". So for example, p(1) = 1, p(2) = -4, p(3) = 1, etc. Now consider the cumulative summing function s(n), which is the sum of the first n terms of p. So s(1) = 1, s(2) = -3, s(3) = -2, etc.. My questions are:
1. s(n) > 0 for a few values up to n=13, after that it seems to stick to negative values. My question is, which is the smallest n>13 such that s(n)>0? It has to be pretty big, I tried the first 10 000 digits of pi and didn't reach it. This fact is interesting since the function shouldn't "prefer" negative values over positive ones. But s(n) seems to "get stuck" in the negative numbers and has a hard time getting out of there.
2. Is s(n) bounded above or below? My intuition says no, but I cannot formally prove it.
3. Consider the histogram of the values s(1), s(2), ..., s(n), for ever increasing n. I suspect this should converge to a normal distribution, but again I cannot formally show this. If it does converge to a normal distribution, what are the parameters of said distribution?
4. (bonus question): can you generalize your answers to questions 2 and 3 to any irrational number? For that matter, are the answers to questions 2 and 3 always the same for all irrational numbers?
1. s(n) > 0 for a few values up to n=13, after that it seems to stick to negative values. My question is, which is the smallest n>13 such that s(n)>0? It has to be pretty big, I tried the first 10 000 digits of pi and didn't reach it.
Interesting question! The first large positive s(n) is s(16075) = 2.
I used this script:
edit: fixed the URL
Language: lua
local PI = "too long to display here" -- see http://www.geom.uiuc.edu/~huberty/math5337/groupe/digits.html
local house = 0
local sum = 0
for digit in string.gmatch(PI, "%d") do
digit = tonumber(digit)
house = house + 1
sum = sum + (house%2 == 0 and -1 or 1)*digit
if house > 13 and sum > 0 then
print("Found solution", house, sum)
break
end
end
print"end"
