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anonymous
 4 years ago
Gotta another annoying question.
Last one for the night
anonymous
 4 years ago
Gotta another annoying question. Last one for the night

This Question is Closed

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0This delves into the issue of convergence. Here's something to think about. Does \[ 0.\bar{9} = 1 \] ?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Where I'm using the convention that the bar overtop means "repeating"

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0umm nearly but i wldnt say it equals 1

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Okay. By how much does it differ from one?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0By which I mean, if it's not equal, than how close is it?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0.0000000000000000000000000001

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0it is an infinite number

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0No, it must be closer than that, those nines go on forever.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0If I ask you "What's the difference between 0.9 repeating and 1", could you give me any real number?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0i dont think so but i may be wrong but if I had to give a number i wld say 0

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0That would be reasonable. But if the difference between two numbers is zero, what does that mean?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0They are one in the same

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Indeed they are. Mathematically, 1 and 0.9 repeating represent exactly the same number. A bit nonintuitive at first, I'd imagine.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0But that raises the question, what makes two quantities equal? Debating that 2 = 2 and 8 = 8 seems to be kind of silly, and I wouldn't argue with that. But you must understand that the quantity 2 and the shape that you draw on a piece of paper when you'd like to represent the quantity are not the same thing.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0I could just as easily express the quantity "two" as \[ \sum_{n=0}^\infty \frac{1}{2^n} \]

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0huh didnt understand that part

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0how cld 2 equal that?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0If you haven't studied infinite series then you probably wouldn't. But it does. I promise :)

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0lol ok trust u on that one

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0So this finally leads us to the issue of convergence. When you say that \[\lim_{x \rightarrow \infty} f(x) = y\] What does that mean? In nonrigorous terms, it means that you can make f(x) as close as you want to y by making x larger.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0ok in other words as when u use reimann sums u make many rectangles?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Example: I argue that \[\lim_{x \rightarrow \infty} \space \frac{1}{x} = 0\]. My friend disagrees. His argument is that the equality is not correct.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0lol but like math books wld say it =0

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0So, I would say to my friend what I said to you earlier: Alright, so if the two sides of the equals sign are different, by how much do they differ? And the key to this is that he could not provide me with a real number by which the two sides would differ that I could not disprove simply by making x larger.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0lol so i guess I shld take ur side of the arguement

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0If he said, "oh, well, they differ by one billionth" then I would say "as soon as x becomes larger than a billion, that's not true". I could repeat that argument forever.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Awesome explanation Jemurray3 :)

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0He probably cant sleep cus he is too busy thinking abt math

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Oh no someone is spying on me again. It happens every night. Do u see that guest?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Thank you guys, you are too kind :) But does that help in your understanding of convergence? That particular integral is defined to be \[\lim_{a \rightarrow \infty}\space \int_{1}^a \space \frac{1}{x^2} dx\] That integral is just \[\lim_{a \rightarrow \infty} \space 1\frac{1}{a} \] Which we can make as close as we'd like to 1 by making a larger and larger and larger. That's the real idea behind a limit, and the convergence of improper integrals.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0And you're right about the not sleeping sometimes, my nerdiness is almost depressing ;)

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0lol I got it thanks. That was very clear

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0U always come save me

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0I'm glad, it's a tricky and nonintuitive idea. It seems weird, but then you study it and it seems fine, but then you take a closer look and it gets weird again.... it takes awhile to really get comfortable with it, but I find this particular part of mathematics to be a little gem that is too often skipped over.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0LOL I guess my prof also thinks it is a gem that shldnt be skipped ove thanks gnite
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