Question

Gotta another annoying question.
Last one for the night

Answer 1

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

Answer 2

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

Answer 3

umm nearly but i wldnt say it equals 1

Answer 4

ya i know that

Answer 5

Okay. By how much does it differ from one?

Answer 6

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

Answer 7

.0000000000000000000000000001

Answer 8

it is an infinite number

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

They are one in the same

Answer 14

no difference lol

Answer 15

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

Answer 16

yes

Answer 17

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.

Answer 18

ok

Answer 19

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

Answer 20

huh didnt understand that part

Answer 21

how cld 2 equal that?

Answer 22

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

Answer 23

lol ok trust u on that one

Answer 24

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.

Answer 25

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

Answer 26

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.

Answer 27

lol but like math books wld say it =0

Answer 28

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.

Answer 29

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

Answer 30

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.

Answer 31

Awesome explanation Jemurray3 :)

Answer 32

jemurray is awesome

Answer 33

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

Answer 34

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

Answer 35

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.

Answer 36

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

Answer 37

lol I got it thanks. That was very clear

Answer 38

U always come save me

Answer 39

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.

Answer 40

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