https://www2.kenyon.edu/Depts/Math/Paquin/GeomSeriesCalcB.pdf

Here’s a standard introduction to the concept of the Convergence/Divergence Theorem of Geometric Series, starts on page 2.

Its quite common for this to be referred to as the convergence test or rule or theorem by teachers and TA’s.

The ellipsis notation generally refers to repetition of a pattern.

Ok. In mathematical notation/context, it is more specific, as I outlined.

This technicality is often brushed over or over simplified by math teachers and courses until or unless you take some more advanced courses.

Context matters, here’s an example:

Generally, pdf denotes the file format specific to adobe reader, while in the context of many modern online videos/discussions, it has become a colloquialism to be able to discuss (accused or confirmed) pedophiles and be able to avoid censorship or demonetization.

0.999… is a real number, and not any object that can be said to converge. It is exactly 1.

Ok. Never said 0.999… is not a real number. Yep, it is exactly 1 because solving the equation it truly represents, a geometric series, results in 1. This solution is obtained using what is called the convergence theorem or rule, as I outlined.

In what way is it distinct?

0.424242… solved via the convergence theorem simply results in itself, as represented in mathematical nomenclature.

0.999… does not again result in 0.999…, but results to 1, a notably different representation that causes the entire discussion in this thread.

And what is a ‘repeating number’? Did you mean ‘repeating decimal’?

I meant what I said: “know patterns of repeating numbers after the decimal point.”

Perhaps I should have also clarified known finite patterns to further emphasize the difference between rational and irrational numbers.

EDIT: You used a valid and even more mathematically esoteric method to demonstrate the same thing I demonstrated elsewhere in this thread, I have no idea why you are taking issue with what I’ve said.

Pack 100 of compsognathus (compsognathii?) says hello.

Not sure how out of date the research is, but in the original Jurassic Park book, there are roaming packs of these things that overwhelm and kill people.

Though the on screen scene of them killing people happens in the second movie, it actually takes place in the first book IIRC… anyway, they’re basically depicted as land piranhas.

(Again, IIRC, Jurassic Park the book basically gets set in motion with a family of tourists being eviscerated by a pack of compys… but the first movie dropped this from the story, then when the second movie comes out they basically use this scene as the intro for that, but its on a different island and used to set off an entirely new story?)

There are a lot of concepts in mathematics which do not have good real world analogues.

i, the _imaginary number_for figuring out roots, as one example.

I am fairly certain you cannot actually do the mathematics to predict or approximate the size of an atom or subatomic particle without using complex algebra involving i.

It’s been a while since I watched the entire series Leonard Susskind has up on youtube explaining the basics of the actual math for quantum mechanics, but yeah I am fairly sure it involves complex numbers.

The explanation I’ve seen is that … is notation for something that can be otherwise represented as sums of infinite series.

In the case of 0.999…, it can be shown to converge toward 1 with the convergence rule for geometric series.

If |r| < 1, then:

ar + ar² + ar³ + … = ar / (1 - r)

Thus:

0.999… = 9(1/10) + 9(1/10)² + 9(1/10)³ + …

= 9(1/10) / (1 - 1/10)

= (9/10) / (9/10)

= 1

Just for fun, let’s try 0.424242…

0.424242… = 42(1/100) + 42(1/100)² + 42(1/100)³

= 42(1/100) / (1 - 1/100)

= (42/100) / (99/100)

= 42/99

= 0.424242…

So there you go, nothing gained from that other than seeing that 0.999… is distinct from other known patterns of repeating numbers after the decimal point.

Yeah, its infuriating that punk has become a suffix.

There is nothing punk about steampunk, dieselpunk, atompunk. They are just fantasy technological scenarios / art styles.

Cyberpunk has an both a recognizable aesthetic and a whole lot of political, social and philosophical views baked into it. You get the punks in cyberpunk as either a direct ideological opposition to the power of corporations, or as an indirect result of said corpos creating a hell world for 99% of people.

There is nothing inherently rebellious about worlds or characters within worlds with more prevalent / advanced steam or diesel or nuclear power.

Solarpunk arguably has some actual punk to it if you actually try to follow the idea of personally minimizing your fossil fuel usage, but mostly its a utopian or post-dystopian setting / art style.

Its now like -gate being affixed to any kind of publicized controversy.

Most people do not understand what Watergate even was and why it was so significant.

Nah, it was my handwriting, used the same lingo and joke I remember using at the time, took up most of the board… which is why I was so shocked it hadn’t been erased.

Most of the time people wrote on it, they were gracious with the space.

Due to typing far more often than writing, and many years later me figuring out oh haha I’m actually naturally left handed, my writing is a fairly uncommon kind of small cap block print, otherwise I am basically the only one capable of reading my non block print scribbles.

I am not quite that old, but I learned digital media creation in the Macromedia Suite… Dreamweaver, Flash… before Adobe bought them out.

I recently returned to a hole in the wall restaurant that I used to frequent in my college days. One of the few actual restaurants in the city that never closed.

A decade later, a semi nonsensical scrawl about being kind and good to people I’d written on a dry erase board while quite drunk… was still there.

It only needed minor updating to be more gender inclusive, which someone else had done without removing any of my writing.

When I was frequenting this restaurant, the whole board was wiped every week or so.

For some reason, what I wrote persisted for a decade.

I couldn’t believe it.

I thought Paulogia’s minimum witnesses argument is basically that Paul could have hallucinated, and that those who witnessed an empty tomb basically did see an empty tomb, but circumstantial confusion led them to misinterpret what they saw?

I’ll have to rewatch some of his vids.

Also, hey, Goju Ryu! I trained in Shito Ryu =D

All things are possible through Christ lol

Its generally more up to date with newer standards and such than Debian, but it is by no means bleeding edge.

Bleeding edge is generally bad unless you really need some specific thing for a specific reason.

If your whole set up is bleeding edge then congrats, you are a basically alpha testing an OS.

Yeah, Arch is not a great choice.

I would suggest PopOS! or Nobara.

Or just good ole reliable Debian.

Ya’ll need God.

Posting this from TempleOS

I like this answer too!

This stems from the fact that, so far, the earliest dated written fragments we have from what is now the New Testament are some of the writings of Paul.

Paul was not one of the Apostles EDIT: Disciples, and it seems possible that, after persecuting earlier, existing Christians, he could have basically had a stress induced psychotic break and hallucinated the vision of Jesus that he had, then converted.

Thing is though, Christians would have to … you know exist and already be a real thing first, for that to make sense.

It does explain why Paul does not mention some very key elements of the narrative of the Gospels: He just had not actually read about or heard of those parts yet.

This creates some theological problems down the line, and some of those problems were ‘remedied’ by what a good deal of scholars and historians believe to be forgeries… chapters of the Bible that modern Christians attribute to Paul, but do not seem to actually have been written by Paul.

It is also possible to some of the empty tomb accounts in some of the Gospels as similar kinds of trauma induced hallucinations.

Mark famously originally just ends with an empty tomb, and nobody said anything about this because they were scared… and then the last bit of verses giving Mark a more satisfying ending have been shown to be added … decades later.

There exists documented proof in many bits of literature from around 200 BCE to around 100 CE of numerous different figures in what is called ‘Jewish Apocalypticism’, basically a small in number but persistent phenomenon of Jews in and around what was for most of that time the Roman province of Palestine, preaching that the end would come, that God or a Messiah would return or arise and basically liberate the region and install a Godly Kingdom, usually after or as part of other fantastical events.

Jesus was one of many of these Jewish Apocalypticists. Much like the rest of the movement’s key figures, they were wrong, and their lives were greatly exaggerated in either their writings or writings about them or inspired by them.

This seems to be the (extremely condensed) opinion of most Biblical Scholars.

There are a very small number of modern Biblical Scholars that are ‘Mythicists’ of some kind, who believe that Jesus was completely fictional and wholly invented by certain people or groups.

This is an unpopular view amongst scholars and historians of that time and region, as most believe it more plausible that Jesus was just another example of a radical Jewish Apocalyptic preacher, which again, was fairly common for roughly 300 years in that region.

Its like how if you go to a big city theres always that one guy with a megaphone preaching imminent doom. 99% of people think this is silly and ignore them, but tons of people know that people like them exist and do have small followings.

Depends on how the combo works. Is there an element of skill involved? If it’s like a rhythm game I would just call it a puzzle/rhythm game. Otherwise it’s just a puzzle game with extra steps.

That was in my OP though, that most games can be thought of as puzzle games with extra steps.

EDIT: I would agree that Tetris is not a puzzle game.

This is quite interesting to me.

My main point of this post is to highlight how the genres and categories we have for games breaks down upon examination and I guess changes over time.

I would say that most people either have or would call Tetris basically the most popular puzzle game.

When it came out it was basically the titular, archetypal ‘puzzle’ game.

You think that pushing a button that generates a purely random outcome is a game?

To me, those are neither games nor puzzles.

There is nothing one can do, in terms of thought or execution, to influence the outcome.

Other than I suppose choosing to play or not play.

To me, a game must include some capacity of the player to influence whether they succeed or fail, within the game itself.

I will grant you that, I agree.

If there is no thought required to determine the difference between a correct and incorrect choice, if its purely just ‘do this’ and the only difficulty is in execution, then yes I would agree this is not a puzzle.