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

1. anonymous

2. anonymous

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

3. anonymous

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

4. anonymous

umm nearly but i wldnt say it equals 1

5. anonymous

ya i know that

6. anonymous

Okay. By how much does it differ from one?

7. anonymous

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

8. anonymous

.0000000000000000000000000001

9. anonymous

it is an infinite number

10. anonymous

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

11. anonymous

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

12. anonymous

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

13. anonymous

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

14. anonymous

They are one in the same

15. anonymous

no difference lol

16. anonymous

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

17. anonymous

yes

18. 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.

19. anonymous

ok

20. anonymous

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

21. anonymous

huh didnt understand that part

22. anonymous

how cld 2 equal that?

23. anonymous

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

24. anonymous

lol ok trust u on that one

25. 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.

26. anonymous

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

27. 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.

28. anonymous

lol but like math books wld say it =0

29. 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.

30. anonymous

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

31. 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.

32. anonymous

Awesome explanation Jemurray3 :)

33. anonymous

jemurray is awesome

34. anonymous

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

35. anonymous

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

36. 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.

37. anonymous

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

38. anonymous

lol I got it thanks. That was very clear

39. anonymous

U always come save me

40. 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.

41. anonymous

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