Whose credibility? (I believe the answer is yours.)
And why should I care? (I believe that I shouldn’t.)
You are missing the point. There are sequences that aren’t algorithms. (\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dotso) is an example of a sequence that isn’t an algorithm.
Algorithms are (some algorithms, that is) specific type of sequences, namely, sequences of instructions on how to perform certain task. Most algorithms, however, are closer to being trees than sequences.
That’s not an algorithm. That’s a sequence of mathematical expressions (written poorly.)
1 = 1
1 = 2 x 1/2
1 = 4 x 1/4
1 = 8 x 1/8
1 = 16 x 1/16
1 = 128 x 1/128
etc
What do you mean when you say that the other side (the right side, I guess) converges to zero? The right side is at all times equal to 1.
That’s a sequence of instructions (an algorithm), whereas (1, 2, 3, \dotso) is a sequence of numbers (not an algorithm.)
I have no idea what you’re talking about. Unless you make an effort to clarify your position, this is the end of the discussion between the two of us. You have no right to complain at this point.
(\frac{1}{8}) is NOT the right side of that equation, it’s merely one small part of it. The right side is (\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}) and it’s equal to (1).
And that’s why you lost credibility in this thread. Like I stated before. You have your fancy symbols… but plain English is something you fail to grasp.
I’m not speaking this in complex ways.
I wish I could dumb it down for you, but I’m not sure that I can.
I’ll try again…
1=1/1
1=0.5+0.5
1=0.25+0.25+0.25+0.25
Etc…
On the right side, the numbers get lower and lower and lower.
For example: the next step is:
1=1/8+1/8+1/8+1/8+1/8+1/8+1/8+1/8
…
Eventually, in this series, if limits converge, every number to the right will equal zero.
I honestly cannot make it simpler than that. I’m sorry, it’s impossible to me to make it simpler!
I complain because we live in a zero sum world. Where there’s a winner, there’s a loser.
I complain about my losses and I complain about my victories. I hate this world. If you had any sense, could put just 2 neurons together in that brain of yours, you’d hate it too.
It makes perfect sense and I understand the argument, and you’ve explained it fine for anyone with a brain.
The limit of each number on the right is indeed 0, as all the denominators tend towards infinity. Add all those limits together to get 0, when the left hand side remains 1 throughout.
The problem is that you have an infinite number of fractions on the right hand side.
This undefined element is what you’re not respecting, because you can’t only pay attention to “any number of zeroes added together is zero”. You also have to pay attention to “an infinite number of positive numbers added together is infinite” and also “any number of that same number’s reciprocal added together is one” etc.
1=“0” isn’t the only answer you can get, but it’s the only answer you’re paying attention to.
I’m not being biased when I say “look at all the possible solutions, not just one”. That’s literally the opposite of being biased.
I’m sorry, but it’s just not a valid proof that there’s something wrong with math. Nice try, but no cigar.
What’s interesting to me here silhouette, is that if this series doesn’t converge at zero (as you state) then 0.9… cannot equal 1. You’re arguing against yourself here!
I’m creating a very specific box with this proof:
If my proof is false, YOU’RE the one destroying math, not me!
Math goes on just fine if convergence is false! If convergence is true … that’s the end of all math!
if the fractions do actually reach their limit of 0, then the right hand side properly equals 0 when the left hand side still equals 1, and the difference between (0.\dot9) and (1) can also genuinely reach its limit of 0, but math is therefore broken? And,
if the fractions on the right hand side of your equations don’t reach their limit of 0, leaving something to work with to maintain equality with the left hand side of 1, then there is the same kind of “something” between (0.\dot9) and (1), and mathematical consistency is maintained?
You let me know if I understand the message.
What I’m trying to say is that those aren’t the only two options.
The limit of the equations you’re presenting leads to an undefined number of fractions, such that whether each fraction reaches their limit of 0 or not, the right hand side of the equation isn’t neatly “0” (or anything else) until you do more work and fully narrow down a singular defined answer only.
By contrast, there’s no extra work to do to get the difference between (0.\dot9) and (1). There is no other number that it can be than the singular defined answer of zero. You could arbitrarily phrase the difference as (\lim_{n\to\infty}(n\times\frac1n)=0) just like in your argument to try and force the same complication, because the the difference may as well be (\sum_{n=0}^\infty\frac0{10^n}=0), but we already know the singular defined answer never gets to a point where it’s larger than 0 no matter how far you go down the decimal expansion.
It’s simply not the same situation, so the complication you stop at with your argument, instead of working it through to get beyond the undefined element and finding a way to fully get to a singular defined answer, doesn’t apply. It doesn’t prove anything by itself, never mind anything about (1=0.\dot9), because you stopped short and concluded a singular defined answer without doing more work to resolve the undefined element that could just as easily yield a different answer for you to stop short at instead, if you wanted.
Let me know if you, in turn, understand my message.
Not the limit of every number on the right side (that makes no sense) but the limit of the sequence of numbers that is formed by picking one number from the right side of every equation. That sequence is (\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dotso) and its limit is indeed (0).
Why are we adding those limits together?
Let’s say that his starting assumption is that an infinite sum of (\lim_{n\to\infty} \frac{1}{n}) is equal to (1) (even though that doesn’t appear to be the starting point of his argument) then what he is saying is the following:
The conclusion logically follows from the premises.
But since we know the conclusion is wrong, it follows that some of our premises are wrong. Ecmandu is claiming that the premise that is false is the premise that infinite series converge. I disagree. What’s actually wrong is the premise that (1) is equal to an infinite sum of limits of the sequence (\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dotso).
Mathematician:
Non-mathematician: <no questions asked, just straight to…> “no, that makes no sense”.
Yes it does make sense. The number that’s the outcome of each term on the right hand side tends towards a limit of 0 as the denominator tends towards infinity. This is a fact, so just stop with the premature ejaculation and for once try and figure out why it’s a fact first before blurting out that it doesn’t yet make sense TO YOU and is therefore wrong.
If you mean the sequence of each fraction on the right hand side, then yes - that’s what I literally just said.
If you mean those are the terms on the right hand side of the equation, then no - that’s not what he said.
Either way, shh, listen, learn, and stop pretending.
Because that’s literally the pattern he’s trying to communicate and the whole point of his argument maybe?
It is an incomplete part of the starting point of his argument.
Multiply that limit by “n”, and it equals 1.
This is clearly the case for all finite values of “n” on the way to infinity, but not so clear for when “n” is infinite.
The logic is incomplete, as I literally just described.
As I literally just described there are various patterns to use to get to various different conclusions.
The premises are fine, the incomplete logic is wrong. As I literally just described.
He’s not even saying that this sequence that you keep repeating is 1.
And infinite series absolutely can converge (either that or they diverge).
Why do you keep talking to me when you keep complaining that I keep calling you out on your bullshit? What did your post just achieve? It’s not even clear you understand his simple argument from several things that you’re saying and you’re still asserting your conclusions as fact without a hint of self-questioning. Some things you’re “telling me” when I literally said them just before. And the one question you actually did ask, I literally just answered it in my last post.
Like seriously, what the fuck was that post of yours? A waste of everyone’s time, STILL no indication whatsoever that you’ve adjusted your approach in the slightest to reflect the reality of your expertise on the subject, or that you’ve learned a single thing, only an attempt to talk to me again to hear what I keep telling you you need to do before this whole thing can even begin to be remotely constructive.
