Gotta another annoying question.
Last one for the night

- anonymous

Gotta another annoying question.
Last one for the night

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- anonymous

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- anonymous

This delves into the issue of convergence. Here's something to think about. Does
\[ 0.\bar{9} = 1 \]
?

- anonymous

Where I'm using the convention that the bar overtop means "repeating"

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- anonymous

umm nearly but i wldnt say it equals 1

- anonymous

ya i know that

- anonymous

Okay. By how much does it differ from one?

- anonymous

By which I mean, if it's not equal, than how close is it?

- anonymous

.0000000000000000000000000001

- anonymous

it is an infinite number

- anonymous

No, it must be closer than that, those nines go on forever.

- anonymous

If I ask you "What's the difference between 0.9 repeating and 1", could you give me any real number?

- anonymous

i dont think so but i may be wrong but if I had to give a number i wld say 0

- anonymous

That would be reasonable. But if the difference between two numbers is zero, what does that mean?

- anonymous

They are one in the same

- anonymous

no difference lol

- anonymous

Indeed they are. Mathematically, 1 and 0.9 repeating represent exactly the same number. A bit non-intuitive at first, I'd imagine.

- anonymous

yes

- anonymous

But 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

ok

- anonymous

I could just as easily express the quantity "two" as
\[ \sum_{n=0}^\infty \frac{1}{2^n} \]

- anonymous

huh didnt understand that part

- anonymous

how cld 2 equal that?

- anonymous

If you haven't studied infinite series then you probably wouldn't. But it does. I promise :)

- anonymous

lol ok trust u on that one

- anonymous

So 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 non-rigorous terms, it means that you can make f(x) as close as you want to y by making x larger.

- anonymous

ok in other words as when u use reimann sums u make many rectangles?

- anonymous

Example: 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

lol but like math books wld say it =0

- anonymous

So, 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

lol so i guess I shld take ur side of the arguement

- anonymous

If 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

Awesome explanation Jemurray3 :)

- anonymous

jemurray is awesome

- anonymous

He probably cant sleep cus he is too busy thinking abt math

- anonymous

Oh no someone is spying on me again. It happens every night. Do u see that guest?

- anonymous

Thank 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

And you're right about the not sleeping sometimes, my nerdiness is almost depressing ;)

- anonymous

lol I got it thanks. That was very clear

- anonymous

U always come save me

- anonymous

I'm glad, it's a tricky and non-intuitive 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

LOL I guess my prof also thinks it is a gem that shldnt be skipped ove
thanks gnite

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