# 1=0.999... infinities and box of chocolates..Phliosophy of Math...

Hansda, consider $$0.999... = \sum_{k\geq 1} \frac{9}{10^k}$$
When k = infinity(consider it(k) as hyper-real or extended real number), how do you think the infinity-th term would be in the RHS of the above equation?

If you want to stop at $$k = \infty$$ you have three problems. $$\infty$$ is not a counting number (not any type of number in standard practice), the expression you are quoting doesn't have an upper bound on where to stop, so you want to talk about $$S_{\infty}$$ if $$\infty$$ was a counting number, and since if $$\infty$$ was a counting number then $$S_{\infty} = 1 - 10^{-\infty}$$ which it stands to reason that: $$1 - \frac{1}{\infty} \lt 1 - \frac{1}{\infty^3}\lt S_{\infty} \lt S_{\infty+1} \lt S$$ (because this is true for all counting numbers) but the precise value of $$S_{\infty}$$ and how it relates to $$S_{\infty-1}$$, $$S_{\infty+1}$$ and $$S$$ requires you to do the heavy lifting. Specifically, I do not know that the hyperreals really have a good handle on exponentials of infinity quantities because non-standard analysis is a topic for math graduate students for the most part, but I thought this was one of the problems that they had.

In the hyperreals, $$\infty$$ is not the largest possible number -- there is no such number. In the hyperreals, $$\infty-1 \lt \infty \lt \infty + 1 \lt \infty^2 \lt \infty^3 \lt e^{\infty} \lt 10^{\infty}$$ because $$e^{\infty} = 1 + \infty + \frac{\infty^2}{2!} + \frac{\infty^3}{3!} + \dots$$ but I don't know how they get detailed answers from that.

In the extended real numbers $$\infty$$ doesn't have any properties of a counting number, such that $$\infty-1 \not\lt \infty \not\lt \infty + 1$$ so specifically $$S_{\infty} = S = 1$$.

Because the concepts of infinity are different, if you want answers you have to do the heavy lifting.

Undefined:

I asked you whether you consider the mathematical statement 1/2 + 1/4 + 1/8 + ... = 1 to be true or false.

Want to try again, without all the off-topic waffle?

Actually, it turns out that even Trippy was eventually moved to have reference to and invoke the 'reality' when the 'abstract/math' approach proved trivial and self-referencing in its circuity and non-independent 'proofing' arguments. That is why these discussions must immediately/ultimately have reference to reality, if they are not to just drift off into the abstractions 'ether' from whence there is no escape or return if one is not careful!

Off-topic? Not in the context of what I and others have been discussing (and pointing out; see my reply to arfa above as an example) which confirms the basically philosophical (not mathematically logical a priori) nature of the starting premises (like 'dimensionless point' notions etc.) of the very mathematical Axioms themselves. I also pointed out to arfa that the first UNITARY entity (real, not abstract philosophy) is ultimately based on the "infinitesimal quantum of minimal effectiveness" FROM WHICH all other 'constructions' (both physical and mathematical) numbers/values arise and revert to via the evolutionary/devolutionary processes involved/applied by nature/mathematics.

Those FRACTIONS (ie, 1/2 + 1/4 + 1/8 + ...) you wrote down are only notation symbols from convention, and NOT actual process. Hence the constructions/deconstructions always relate back to UNITY (that "1" notation convention symbol), hence ANY fraction that ipso facto CAME ORIGINALLY and TRIVIALLY from that 'unity', then must equally trivially return/reconstruct back to that unity.

See? There is nothing to 'prove', either way, by such trivial manipulations/notations which have not actually IDENTIFIED the real/logical process. Such notation expressions are not actual ACTIONS/OPERATIONS 'already done', they are expectations of things YET TO BE DONE....but NOT ACTUALLY IDENTIFIED or proven in fact YET. So the expression may have 'mathematical overtones', but it is meaningless for any explanatory/proofing of things not yet done/settled. The "1" is always "1"; and the "fractions are always "fractions"; but the static statements/equivalences etc do not represent the real essence of what these are.

That is why I cannot 'buy into' that obviously trivial exercise as anything that will lead to anything 'proven'. Hence my previous reply advising that there is more to it than what your question implies. For the whole idea of what I and others have been getting at, you'll just have to wait until my math-physical ToE 'combo' is out. Until then I can't say any more for fear of plagiarizers (not by the honorable Sciforum members, naturally! ). That is an unfortunate fact of life which I have to be cognizant of when attempting to reply as fully as I can without letting too many cats out of the bag prematurely. You understand! Sorry, that's all the participation/replies I can do at this time. Thanks again for your polite and interesting inputs, James.

Well. that all looks impressive and all, but what about counting sheep or camels?
Can we say a camel is "an infintesimal of physical effectiveness", or "a real indivisible state"?

I mean, humans started using numbers by counting whole things and keeping records, thereby "inventing" number systems. The notational conventions of the day were sufficient or fit for their purpose, and they survived as long as they remained so (viz the persistence of the Roman number system and its eventual replacement with the Arabic system).

Now you propose to "update" our concept of numbers, but haven't really pointed out why they are no longer fit for the purpose. We have computers that seem to do quite a good job handling numbers (as digital on/off states), I just can't see why, or even how, this needs revising. Computers of course, can't handle infinities because they all have an "overflow problem". Infinity is a concept though; computers don't "handle" concepts.

Apart from the indivisible minimal infinitesimal of effectiveness QUANTUM ENTITY, all else is "physical/mathematical logic CONSTRUCTION and DE-construction from/back to that entity. Until you 'get' what the implications for both maths and physical logical manipulations and symbolic notations, you will miss the essential insight which has already answered your own question.

When you LABEL a collection of quanta as 'constructing a CAMEL, it is YOU making the philosophical identity FOR such a 'thing' which you label "1" Camel, "2" Camels individually OR use a LOCALIZED COLLECTIVE TERM and call MANY Camels "1" HERD, "2" Herds of Camels.

See?....you have to distinguish the fundamentals from the overlays which you and maths get lost in and end up with "undefined", "undetermined" etc etc which flow from 'dimensionless point' and other UNREAL 'things'.

Anyhow, I have pointed out all I can point out at this time. My Math-Physics ToE 'combo' will have to disclose the full explanation consistent and complete. Sorry if the clues and hints and examples and observations so far here and elsewhere haven't resulted in moving you away even a little from abstractions towards reality maths/physics approach/understandings/insights. If so, you'll just have to wait for the rest, along with everybody else, mate! Cheers...and sincere congratulations again, arfa!

Undefined said:
When you LABEL a collection of quanta as 'constructing a CAMEL, it is YOU making the philosophical identity FOR such a 'thing' which you label "1" Camel, "2" Camels individually OR use a LOCALIZED COLLECTIVE TERM and call MANY Camels "1" HERD, "2" Herds of Camels.
Look, a man sees some camels and counts them. Why would they think each animal is a "collection of quanta constructing a camel"? Isn't that like, completely uneccessary?
See?....you have to distinguish the fundamentals from the overlays which you and maths get lost in and end up with "undefined", "undetermined" etc etc which flow from 'dimensionless point' and other UNREAL 'things'.
How do I get lost counting camels (which by the way is something humans have been doing for thousands of years)? Counting is fundamental; there is no "undefined" or "dimensionless point", although the numbers as labels do not have dimensions, but camels do. What's the problem?
Anyhow, I have pointed out all I can point out at this time. My Math-Physics ToE 'combo' will have to disclose the full explanation consistent and complete. Sorry if the clues and hints and examples and observations so far here and elsewhere haven't resulted in moving you away even a little from abstractions towards reality maths/physics approach/understandings/insights.
You haven't pointed out anything, all you've done is insist there are problems with mathematical abstractions, which appears to be confined to them being abstractions. Numbers always have and always will be abstract, so I really can't follow your argument.

As for "reality maths", there is an awful lot of mathematics that appears to have no representation in the physical world. What do you think you can do about that, and the fact that the whole of physics is described by a rather small set of symmetries and relations between sets, whereas mathematics is not constrained this way?
Have you heard of the Mandelbrot set? Do you think it's a mathematical reality even though it isn't physical--that is, we don't "observe" it?

Undefined said:
Those FRACTIONS (ie, 1/2 + 1/4 + 1/8 + ...) you wrote down are only notation symbols from convention, and NOT actual process.
Mathematics is notation, you dork.

When you write a series like a + b + c + ..., it doesn't describe a process either. The process of adding terms together is something we do, it's implied by the "+" symbol.
See? There is nothing to 'prove', either way, by such trivial manipulations/notations which have not actually IDENTIFIED the real/logical process.
Then what do we use, if not mathematical notation, to prove anything about numbers?
Such notation expressions are not actual ACTIONS/OPERATIONS 'already done', they are expectations of things YET TO BE DONE....but NOT ACTUALLY IDENTIFIED or proven in fact YET.
Yeah, usually when you want to prove something it involves manupulations of symbols, substitutions and so on.
That is why I cannot 'buy into' that obviously trivial exercise as anything that will lead to anything 'proven'.
That looks a lot like someone finding an excuse to not bother doing something. There is a way to prove that 1/2 + 1/4 + 1/8 + ... = 1 is true, so it does lead to something proven despite what you say.

And what I'm going to say now is, I'm getting bored with this. So ... bye.

Look, a man sees some camels and counts them. Why would they think each animal is a "collection of quanta constructing a camel"? Isn't that like, completely uneccessary?
How do I get lost counting camels (which by the way is something humans have been doing for thousands of years)? Counting is fundamental; there is no "undefined" or "dimensionless point", although the numbers as labels do not have dimensions, but camels do. What's the problem?You haven't pointed out anything, all you've done is insist there are problems with mathematical abstractions, which appears to be confined to them being abstractions. Numbers always have and always will be abstract, so I really can't follow your argument.

As for "reality maths", there is an awful lot of mathematics that appears to have no representation in the physical world. What do you think you can do about that, and the fact that the whole of physics is described by a rather small set of symmetries and relations between sets, whereas mathematics is not constrained this way?
Have you heard of the Mandelbrot set? Do you think it's a mathematical reality even though it isn't physical--that is, we don't "observe" it?

You missed the point. The labels/names etc you ascribe TO 'things' is NOT necessarily what THE 'thing' IS actually. It is an ASSUMPTIVE description/definition as ONE or TWO 'individual constituent thing', or ONE or TWO 'composite herd of individual things'.

Just because mankind and mathematicians have been 'doing it for thousands of years' is no argument as to its real meanings/origins FOR doing it like that. Especially if, as I have pointed out more than once for your benefit, that what we 'have been doing' is based/follows on UNreal things like dimensionless points, remember?

Did you not even take the time and effort to even TRY to understand what the difference is between a 'dimensionless point' NOTION from philosophy; and a minimal QUANTUM indivisible unitary concept/value o "infinitesimal of effectiveness' from which ALL else MUST follow if more abstractions and philosophy are not to riddle BOTH the maths and the physics and understandings from both?

What maths (and physics too) is NOW, and what maths (and physics too) COULD HAVE BEEN, if that original ancient scholar did not start the UNreal philosophical based NOTION off as some sort of AXIOMATIC basis, is the very question that a REVIEW based on more reality referential axioms and postulates is ALL ABOUT tackling.

Your insistence on what mathematics/physics is NOW and baulking on considering what it COULD be is holding you back and making you angry but no wiser about the new issues being tackled to make both Maths and physics more real FROM THE STARTING AXIOMS/POSTULATES. That is the only way we are going to see what BETTER things BOTH can accomplish and complete in concert.

If you don't want to be part of that reality based review, no sweat. But please don't get angry at others who do want to. Ok, mate?

Mathematics is notation, you dork.

When you write a series like a + b + c + ..., it doesn't describe a process either. The process of adding terms together is something we do, it's implied by the "+" symbol.Then what do we use, if not mathematical notation, to prove anything about numbers?Yeah, usually when you want to prove something it involves manupulations of symbols, substitutions and so on. That looks a lot like someone finding an excuse to not bother doing something. There is a way to prove that 1/2 + 1/4 + 1/8 + ... = 1 is true, so it does lead to something proven despite what you say.

And what I'm going to say now is, I'm getting bored with this. So ... bye.

Language, bro! There are ladies present. It's that sort of language that forces one to respond, even if only to defend against it, hey?

We use the symbols to indicate what the concepts are, not the things themselves.

We use the notations like 1/3 to indicate some sort of implied operation but that is not yet done. No harm in that, sure. But when maths persons attempt to use just those bare expressions/notations as if they 'prove' something as if they were 'arguments' in themselves, then that is what I pointed out is the problem that no such 'proofing' can be had from those trivial notation/symbol manipulations/expressions, That's all. No more than that point. Ok?

To prove anything about numbers we can't properly/logically rely on mere 'more manipulation of maths symbols/notations', we must use REALITY BASED ARGUMENTS predicated on from self-evident REAL truth-axioms about what anything relating to 'unity' and composition/decomposition from/to 'unity' really mean fundamentally. get that yet?

If you start from unreal, [philosophical 'dimensionless point' notions which have no basis in reality BEFORE being co-opted and assumed/subsumed in consequent mathematical constructs as some sort of 'axiomatic truth', then its a slippery slope to where the unreality and undefined and etc 'outputs' we have that indicate that both maths and physics are NOT complete or reality-referential FROM THE START.

Hence the recent push for reality based review 'from scratch' of everything abstract/philosophical currently 'accepted' as maths/physics 'truth' axioms/postulates which, when closely examined turn out NOT to be real things or real 'truths' based at all.

Undefined said:
You missed the point. The labels/names etc you ascribe TO 'things' is NOT necessarily what THE 'thing' IS actually. It is an ASSUMPTIVE description/definition as ONE or TWO 'individual constituent thing', or ONE or TWO 'composite herd of individual things'.
No, I think I managed to get that during my studies, but thanks.
Just because mankind and mathematicians have been 'doing it for thousands of years' is no argument as to its real meanings/origins FOR doing it like that. Especially if, as I have pointed out more than once for your benefit, that what we 'have been doing' is based/follows on UNreal things like dimensionless points, remember?
Tell that to the man counting the camels. I think that man will probably just keep counting.
Did you not even take the time and effort to even TRY to understand what the difference is between a 'dimensionless point' NOTION from philosophy; and a minimal QUANTUM indivisible unitary concept/value o "infinitesimal of effectiveness' from which ALL else MUST follow if more abstractions and philosophy are not to riddle BOTH the maths and the physics and understandings from both?
Well, no. If I have some camels and I want to sell three of them, I'm going to need to do some counting. I won't need to philosophise about minimal effectiveness, since all else (the camels and what I get in return for them) follows from the counting thing.
What maths (and physics too) is NOW, and what maths (and physics too) COULD HAVE BEEN, if that original ancient scholar did not start the UNreal philosophical based NOTION off as some sort of AXIOMATIC basis, is the very question that a REVIEW based on more reality referential axioms and postulates is ALL ABOUT tackling.
What, you mean we shouldn't have started counting things until we were sure about something that seems quite unrelated to the process of counting? Do you sit and think about what your bank balance really means, that it's actually an abstract dimensionless "mathematical" object, a number, or do you just pay the bills?
Your insistence on what mathematics/physics is NOW and baulking on considering what it COULD be is holding you back and making you angry but no wiser about the new issues being tackled to make both Maths and physics more real FROM THE STARTING AXIOMS/POSTULATES. That is the only way we are going to see what BETTER things BOTH can accomplish and complete in concert.

If you don't want to be part of that reality based review, no sweat. But please don't get angry at others who do want to. Ok, mate?
Actually I'm a bit sad. After all that study here I am discussing abstractions with someone who insists abstraction itself is in need of revision. And then there's the expectation that I won't understand any more of your theory, when it arrives, than I've understood of your arguments about what's wrong with mathematics.

Now if you'll excuse me, I have to go and spend some money, or rather, perform an abstract operation on a number.

No, I think I managed to get that during my studies, but thanks.
Tell that to the man counting the camels. I think that man will probably just keep counting.
Well, no. If I have some camels and I want to sell three of them, I'm going to need to do some counting. I won't need to philosophise about minimal effectiveness, since all else (the camels and what I get in return for them) follows from the counting thing.
What, you mean we shouldn't have started counting things until we were sure about something that seems quite unrelated to the process of counting? Do you sit and think about what your bank balance really means, that it's actually an abstract dimensionless "mathematical" object, a number, or do you just pay the bills?
Actually I'm a bit sad. After all that study here I am discussing abstractions with someone who insists abstraction itself is in need of revision. And then there's the expectation that I won't understand any more of your theory, when it arrives, than I've understood of your arguments about what's wrong with mathematics.

Now if you'll excuse me, I have to go and spend some money, or rather, perform an abstract operation on a number.

What's your point about the Camel man and his counting Camels? He is not purporting to engage in arguments and proofs of mathematical axioms and proofs, after all. He is merely using cognitive utility for labeling and keeping simple track of what those labels mean to HIM.

You on the other hand are involved in proofs and axiomatic arguments which purport to explain and justify what/how maths axioms and arguments are used as currently constructed from unreal notion of 'dimensionless point' and other non-essential philosophical abstractions which were unwittingly introduced from the start. I am reviewing everything from that start, and basing the starting and following premises on reality self-evident truths as understood/observed NOW, and not mere purported self-evident abstractions from philosophy of ancient scholars THEN.

No more, no less than that. Your Cameleer is neither here nor there at the level of proof and logics we are at, mate. Good luck with your anger/temper, arfa; you'll never make a great Cameleer' if you don't learn patience and tolerance of 'different individual' types of all kinds, be they Camels or be they other researchers into the reality fundamentals of maths and physics theory/outcomes! No hard feelings I hope. Bye for now to you too, mate.

Undefined:

I've asked you twice now, and you haven't managed to answer the question yet.

1/2 + 1/4 + 1/8 + ... = 1

True or false?

1. 0.5 = 1 / 2 Total = 0.5
2. 0.25 = 1 / 4 Total = 0.75
3. 0.125 = 1 / 8 Total = 0.875
4. 0.0625 = 1 / 16 Total = 0.9375
5. 0.03125 = 1 / 32 Total = 0.96875
6. 0.015625 = 1 / 64 Total = 0.984375
7. 0.0078125 = 1 / 128 Total = 0.9921875
8. 0.00390625 = 1 / 256 Total = 0.99609375
9. 0.001953125 = 1 / 512 Total = 0.998046875
10. 0.0009765625 = 1 / 1024 Total = 0.9990234375
11. 0.00048828125 = 1 / 2048 Total = 0.99951171875
12. 0.000244140625 = 1 / 4096 Total = 0.999755859375
13. 0.0001220703125 = 1 / 8192 Total = 0.9998779296875
14. 0.00006103515625 = 1 / 16384 Total = 0.99993896484375
15. 0.000030517578125 = 1 / 32768 Total = 0.999969482421875
16. 0.0000152587890625 = 1 / 65536 Total = 0.9999847412109375
17. 0.00000762939453125 = 1 / 131072 Total = 0.99999237060546875
18. 0.000003814697265625 = 1 / 262144 Total = 0.999996185302734375
19. 0.0000019073486328125 = 1 / 524288 Total = 0.9999980926513671875
20. 0.00000095367431640625 = 1 / 1048576 Total = 0.99999904632568359375
21. 0.000000476837158203125 = 1 / 2097152 Total = 0.999999523162841796875
22. 0.0000002384185791015625 = 1 / 4194304 Total = 0.9999997615814208984375
23. 0.00000011920928955078125 = 1 / 8388608 Total = 0.99999988079071044921875
24. 0.000000059604644775390625 = 1 / 16777216 Total = 0.999999940395355224609375
25. 0.0000000298023223876953125 = 1 / 33554432 Total = 0.9999999701976776123046875
26. 0.00000001490116119384765625 = 1 / 67108864 Total = 0.99999998509883880615234375
27. 0.000000007450580596923828125 = 1 / 134217728 Total = 0.999999992549419403076171875
28. 0.0000000037252902984619140625 = 1 / 268435456 Total = 0.9999999962747097015380859375
29. 0.0000000018626451492309570312 = 1 / 536870912 Total = 0.9999999981373548507690429687
30. 0.0000000009313225746154785156 = 1 / 1073741824 Total = 0.9999999990686774253845214843
31. 0.0000000004656612873077392578 = 1 / 2147483648 Total = 0.9999999995343387126922607421
32. 0.0000000002328306436538696289 = 1 / 4294967296 Total = 0.9999999997671693563461303710
33. 0.0000000001164153218269348145 = 1 / 8589934592 Total = 0.9999999998835846781730651855
34. 0.0000000000582076609134674072 = 1 / 17179869184 Total = 0.9999999999417923390865325927
35. 0.0000000000291038304567337036 = 1 / 34359738368 Total = 0.9999999999708961695432662963
36. 0.0000000000145519152283668518 = 1 / 68719476736 Total = 0.9999999999854480847716331481
37. 0.0000000000072759576141834259 = 1 / 137438953472 Total = 0.9999999999927240423858165740
38. 0.000000000003637978807091713 = 1 / 274877906944 Total = 0.9999999999963620211929082870
39. 0.0000000000018189894035458565 = 1 / 549755813888 Total = 0.9999999999981810105964541435
40. 0.0000000000009094947017729282 = 1 / 1099511627776 Total = 0.9999999999990905052982270717

looks like 1/2 + 1/4 + 1/8 + ... is going to equal 0.999... ?

looks like 1/2 + 1/4 + 1/8 + ... is going to equal 0.999... ?

Yes, and since 0.999... = 1, it's equal to 1.

0.111... + 0.888... = 0.999...
0.222... + 0.777... = 0.999...
0.333... + 0.666... = 0.999...
0.444... + 0.555... = 0.999...

0.777... + 0.222...3 = 1.000... = 1
0.888... + 0.111...2 = 1.000... = 1
0.999... + 0.000...1 = 1.000... = 1

0.999 + 0.001 = 1
0.999 + 0.00 = 0.999
0.888 + 0.112 = 1
0.888 + 0.111 = 0.999
0.777 + 0.223 = 1
0.777 + 0.222 = 0.999

0.111... + 0.888... = 0.999...
0.222... + 0.777... = 0.999...
0.333... + 0.666... = 0.999...
0.444... + 0.555... = 0.999...

0.777... + 0.222...3 = 1.000... = 1
0.888... + 0.111...2 = 1.000... = 1
0.999... + 0.000...1 = 1.000... = 1

0.999 + 0.001 = 1
0.999 + 0.00 = 0.999
0.888 + 0.112 = 1
0.888 + 0.111 = 0.999
0.777 + 0.223 = 1
0.777 + 0.222 = 0.999
some would argue that 0.999... has a "residual" value lesser magnitude than 0.0...1
that 1 - 0.999... = [a value less than 0.0...1 (an infinitesimal) ] therefore if the residual value is less than an infinitesimal what value must it be?

1 - 0.999... = < (0.0...1]

0.777... + 0.222...3 = 1.000... = 1

If those three dots represent an infinite sequence, then popping a 3 on the end of all those twos is a no no. There's no end to all the twos.

some would argue that 0.999... has a "residual" value lesser magnitude than 0.0...1

You're making the same mistake.

If those three dots mean an infinite series, you can't tack on a "1" at the end, because there is no end.

that 1 - 0.999... = [a value less than 0.0...1 (an infinitesimal) ] therefore if the residual value is less than an infinitesimal what value must it be?

1 - 0.999... = < (0.0...1]

This is meaningless. It was also refuted back in post #915. That post was game over for all arguments that have been put in this thread against the notion that 0.999... = 1.

If you can find a problem with post #915, then you may have an argument worth listening to.

If those three dots represent an infinite sequence, then popping a 3 on the end of all those twos is a no no. There's no end to all the twos.

Yes, its probably a no no, but can you explain why you are doing it when you say 0.999... = 1 ?

hansda:

First thing to say is that a single particle either decays or it doesn't. And there's no way to tell when or if it will, although you can predict on average how long you can expect to have to wait before it decays (based on half-life).

The second thing to say it that if you start with lots of identical particles (say 1 million, for example), then after a few half-lives you can be fairly sure that all will have decayed. You can easily calculate the expected proportion that will have decayed after a certain amount of time; that's what half-life is all about.

So, let's consider an example at random. If you have a lump of radioactive caesium-137, then after 1 half-life, approximately 1/2 of it will have decayed. Wait another half life and 3/4 of it will have decayed. And so on. Wait long enough and all of it will have decayed, on average.

May be you are right but as per wiki and this site, it takes infinite time for complete decay.

May be you are right but as per wiki and this site, it takes infinite time for complete decay.
I think you are misinterpreting the notation:
• $$\lim_{t\to \infty} N(t) = 0$$, i.e. amount approaches zero as ''t'' approaches infinity as expected (the longer we wait, the less remains).
http://en.wikipedia.org/wiki/Half-life#Formulas_for_half-life_in_exponential_decay

Likewise with Yahoo answers, you can't get down to $$10^{-100} \, \textrm{g}$$ of Cs-137 because 1 atom of Cs-137 masses about $$2.2734 \times 10^{-22} \, \textrm{grams}$$.
Eventually you get down to just one atom which will decay at some finite time in the future, but when exactly that is cannot be determined. The longer you wait, the more likely it is to happen. But it very likely will happen in the next 10 half-lives.

If you want to stop at $$k = \infty$$ you have three problems.

Let us stop at k = infinity and try to find the infinity-th term of this infinite geometric series: 0.9 + 0.09 + 0.009 + 0.0009 + 0.00009 + 0.000009 + ... upto infinite terms.

Can you explain about the infinity-th term of this infinite series? What will be the decimal format of this infinite-th term? or do you think that the infinity-th term of this infinite series does not exist?

As per wiki
wiki said:
In mathematics, "infinity" is often treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the real numbers. In number systems incorporating infinitesimals, the reciprocal of an infinitesimal is an infinite number, i.e., a number greater than any real number.

Infinity is also included in hyper real and extended real numbers.