For any general series [i]S(n)[/i] that has a real limit as [i]n[/i] approaches infinity, the limit of its last term is zero.
Basically this means that you can't put infinite amounts of shit into a finite box. It'll overflow. If you want to keep shoveling shit in there ad infinitum, each shovelful needs to be smaller than the last, i.e. the shovelfuls must have a limit of 0 at infinity.
That's the physical explanation, so here's the mathematical one (this first proof is freely copied from my math book):
Defining [i]S(n)[/i] as the sum of its terms.
[img]http://img839.imageshack.us/img839/1301/graph1y.png[/img]
Defining [i]S(n)[/i] to have a real limit [i]L[/i] as [i]n[/i] grows indefinitely. The series is said to converge.
[img]http://img151.imageshack.us/img151/9150/graph2.png[/img]
Any one of its terms, except the first one, can be expressed like this (edit whoops, i > 1 if you start indexing from 1. not like it really matters though):
[img]http://img405.imageshack.us/img405/3365/graph3.png[/img]
Because [i]i[/i] approaches infinity, we can discard the possibility that the term is the first one, and write the limit like this:
[img]http://img225.imageshack.us/img225/999/graph4.png[/img]
IT'S ZERO
That means that whenever there's a series that converges, its terms should have a limit of 0 at infinity.
now on to the second part
[i]f(x) = x - a/2[/i]
[img]http://img837.imageshack.us/img837/1212/ill1.png[/img]
It's obvious from the graph that the integral of [i]f(x)[/i] from 0 to [i]a[/i] always results in 0 because the area below the x-axis is the same as the area above it. Increasing [i]a[/i] scales the graph but doesn't affect the result.
The function [i]f(x)[/i] can be used the same way in a sum.
[img]http://img259.imageshack.us/img259/7376/ill2.png[/img]
We'll define [i]S(a)[/i] as the sum of [i]f(x)[/i] as x goes from 0 to [i]a[/i]. So we're kind of slicing away all the irrelevant parts from the integral; only leaving behind natural values of [i]a[/i].
Because of the symmetry, [i]S(a) = 0[/i] always.
Mathematical definition and proof:
Defining [i]S(n)[/i] as a finite sum of [i](n + 1)[/i] linearly increasing values from [i]-n/2[/i] to [i]n/2[/i]:
[img]http://img221.imageshack.us/img221/7136/graph5.png[/img]
Calculating [i]S(n)[/i] for all natural [i]n[/i]:
[img]http://img88.imageshack.us/img88/9171/graph7.png[/img]
Let's analyze the behaviour of [i]S(n)[/i] as [i]n[/i] grows indefinitely. Since [i]S(n)[/i] doesn't depend on [i]n[/i], its limit is the same old 0. [i]S(n)[/i] "converges" by definition, though it just stays a constant.
[img]http://img440.imageshack.us/img440/5023/graph6.png[/img]
The last term of the series [i]S(n)[/i] is:
[img]http://img88.imageshack.us/img88/690/graph8.png[/img]
The limit of this term as [i]n[/i] grows indefinitely is:
[img]http://img209.imageshack.us/img209/2774/graph9.png[/img]
IT'S INFINITE
The limit of the last term of any series that converges must be 0, as indicated by the first proof, but in the case of at least this one series, it's also infinite.
What's wrong?
This maths :psyduck:
My nose is bleeding.
:psyboom:
To zero and beyond!
By the way, you lost me at Infinity.
I don't know what you're doing there, but infinity is a concept, not a value.
I think you made a mistake in that one part that had a "n" in it.
Based on this simple equation (from wiki), actually, the variables can equal infinity too
[quote]
With the following assumptions:
[IMG]http://upload.wikimedia.org/math/5/9/f/59fbcec15fbbc8744c0a4309c126a8a8.png[/IMG]
The following must be true:
[IMG]http://upload.wikimedia.org/math/d/2/e/d2e283e91dee2cad966314a84da9f1d5.png[/IMG]
Dividing by zero gives:
[IMG]http://upload.wikimedia.org/math/4/6/9/469c83d5e0959e4caaceea20df153c53.png[/IMG]
Simplified, yields:
[IMG]http://upload.wikimedia.org/math/c/4/c/c4c9b852c938da096b69fc257a7a8d82.png[/IMG]
The [URL="http://en.wikipedia.org/wiki/Fallacy"]fallacy[/URL] is the implicit assumption that dividing by 0 is a legitimate operation.[/quote]much easier than the one above too
[QUOTE=A Noobcake;25380295]I think you made a mistake in that one part that had a "n" in it.[/QUOTE]
Yeah, but he TOTALLY fucked up the part with the equal sign.
:psyboom:
These threads, I cannot comprehend.
[QUOTE=Kill001;25380301]Based on this simple equation (from wiki), actually, the variables can equal infinity too
much easier than the one above too[/QUOTE]
I don't think I'm doing anything bad in OP.
[editline]13th October 2010[/editline]
Division by zero is bad
You lost me at 'for'
But infinity = fail mathematically, and zero isn't fail, its zero!
Since when did math involve letters??
Read over what I understood, seems your concept is correct from the top couple of things I've seen so far. If we were to go very fundamental and look at numbers though, 0 is infinite in reality there is no such thing as "terminating decimals." Example. The number 1, it can also be written as 1.0000000000000000000000000000000000000000000000000000000000000000000000000000 etc. You understand the point, at the end of the day it's still 1. 0 is a very powerful number though as we use it everyday. But yeah from what I've read so far, Zero is infinite, not sure if it equals infinite though.
Good tread Puska!
We need more real science/math/physics thread like this one and less psuedoscience. I finished my calculus almost a few months ago. Let me dig up my book again.
[QUOTE=Bledrix;25380475]Since when did math involve letters??[/QUOTE]
:ohdear:
sounds like mathematical bullshit you made up because you're the only nerd who understands it
[QUOTE=ThePuska;25380376]I don't think I'm doing anything bad in OP.[/QUOTE]
No not really, just backing up your claim that zero equals infinity :buddy:
Reminds me of this. Proof that 1+1 does not equal 2.
x = y
-x^2= -xy
x^2 - y^2 = x^2 - xy
(x + y)(x - y) = x(x - y)
x + y = x
1 + 1 =2
2 = 1
[QUOTE=Kill001;25380301]Based on this simple equation (from wiki), actually, the variables can equal infinity too
much easier than the one above too[/QUOTE]
This is bullshit.
Just because the outcome is the same doesnt mean the way you got there is the same.
Fixed the op
[quote=OP]0=Infinity[/quote]
0 is the opposite of infinity don't you know
Infinity is not a number/value.
Sorry for asking and I may be wrong, because I didn't read the whole thing, but...
[quote][url]http://img225.imageshack.us/img225/999/graph4.png[/url][/quote]
But isn't:
lim i => ∞ [S(i)-S(i-1)] = ∞-(∞-1) = ∞-∞
And if I remember my math lessons right, ∞-∞ is not defined in maths, meaning that it cannot be zero.
Then again I did not read the whole thing...
[QUOTE=Vita;25380542]Reminds me of this. Proof that 1+1 does not equal 2.
x = y
-x^2= -xy
x^2 - y^2 = x^2 - xy
(x + y)(x - y) = x(x - y)
x + y = x
1 + 1 =2
2 = 1[/QUOTE]
Step 2 ist right.
[QUOTE=taipan;25380587]This is bullshit.
Just because the outcome is the same doesnt mean the way you got there is the same.[/QUOTE]
Isn't the point of mathematical equations (in this case) is to accurately point out the outcome?
if that was the case, then if you wanted the number 1 off a mathematical equation (for example) then 2-1 or 1+0 works too right? Similar thing here :rolleyes:
[QUOTE=Radman;25380633]Sorry for asking and I may be wrong, because I didn't read the whole thing, but...
But isn't:
lim i => ∞ [S(i)-S(i-1)] = ∞-(∞-1) = ∞-∞
[QUOTE=acidcj;25380616]Infinity is not a number/value.[/QUOTE]
...[/QUOTE]
Yeah this seems to be correct. You are subtracting infinity from infinity which I'm sure you can't really do.
Haven't read the other stuff cause I've forgotten all of it...
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