# 1+2+3+4+ ... equals?

wolfgang59
Posers and Puzzles 06 Oct '10 20:25
1. wolfgang59
Mr. Wolf
06 Oct '10 20:25
1+2+3+4+ ... equals?

ZERO

Obviously 1+2+3+4+ ... = (2-1)+(3-1)+(4-1)+(5-1)+...

=2+3+4+5+... - (1+1+1+1+...)

= inf - inf

= 0
2. Palynka
Upward Spiral
06 Oct '10 20:43
FAIL
3. Palynka
Upward Spiral
06 Oct '10 20:44
Repeat after me "one-to-one correspondence"... "one-to-one correspondence"... "one-to-one correspondence"...
4. wolfgang59
Mr. Wolf
06 Oct '10 21:21
ðŸ˜•

Of course!

I'm trying to point out the flaw in our opponents' argument in the "many balls" thread
5. Palynka
Upward Spiral
06 Oct '10 21:37
Originally posted by wolfgang59
ðŸ˜•

Of course!

I'm trying to point out the flaw in our opponents' argument in the "many balls" thread
Are there more even numbers than natural numbers?

{2} and {1,2}
{2,4} and {1,2,3,4}
...
{2,...,2n} and {1,2,...,n,...,2n}
...
n goes to infinity...
...

The difference in cardinality is always increasing in n!

Yet...the cardinality of the countably infinite sets is the same.

Now repeat after me "one-to-one correspondence"...
6. wolfgang59
Mr. Wolf
06 Oct '10 22:31
Originally posted by Palynka
Are there more even numbers than natural numbers?

{2} and {1,2}
{2,4} and {1,2,3,4}
...
{2,...,2n} and {1,2,...,n,...,2n}
...
n goes to infinity...
...

The difference in cardinality is always increasing in n!

Yet...the cardinality of the countably infinite sets is the same.

Now repeat after me "one-to-one correspondence"...
yes but did you know that there are more evens than odds?

for the odd nmber 1 there is 1+1=2

for the odd nmber 3 there is 3+1=4

for the odd nmber 3 there is 5+1=6

for the odd nmber Nthere is N+1

So for every odd there is an even ... but then we include ZERO

QED There are more evens than odds

ðŸ˜‰
7. 07 Oct '10 04:28
Originally posted by wolfgang59
1+2+3+4+ ... equals?

ZERO

Obviously 1+2+3+4+ ... = (2-1)+(3-1)+(4-1)+(5-1)+...

=2+3+4+5+... [b]-
(1+1+1+1+...)

= inf - inf

= 0[/b]
inf - inf is not zero. It's not even defined. Therefore your calculation is flawed.

If we define inf - inf, inf / inf, and such, as a number then we get funny results, as the one above.
8. 07 Oct '10 06:10
Did you know that 0 = 1?

0 = 0 + 0 + 0 + ...
= (1-1) + (1-1) + (1-1) + ...
= 1 + (-1+1) + (-1+1) + (-1+1) + ....
= 1

QED
9. wolfgang59
Mr. Wolf
07 Oct '10 12:21
Originally posted by FabianFnas
inf - inf is not zero. It's not even defined. Therefore your calculation is flawed.

If we define inf - inf, inf / inf, and such, as a number then we get funny results, as the one above.
I think I knew that!
ðŸ˜•
10. Palynka
Upward Spiral
08 Oct '10 10:21
Originally posted by Thomaster
Did you know that 0 = 1?

0 = 0 + 0 + 0 + ...
= (1-1) + (1-1) + (1-1) + ...
= 1 + (-1+1) + (-1+1) + (-1+1) + ....
= 1

QED
So which standard operation breaks down when you reach an infinite sum? Could you get that result by extrapolating from the union of finite sequences of sums?
11. 08 Oct '10 10:371 edit
If I remember correctly, it's all to do with whether the series is absolutely convergent or not.

SUM(x_i) is absolutely convergent <=> SUM(|x_i|) is convergent.

Absolutely convergent series behave when you rearrange them. Series that are just convergent (e.g. 1 - 1/2 + 1/3 - 1/4 + ...) can converge to different values (or not at all) depending on the order.

For example:
1 - 1/2 + 1/3 - 1/4 + 1/5 +... = ln 2
1 + 1/3 + 1/5 + ..... - 1/2 - 1/4 - 1/6 - ... does not converge

1 - 1 + 1 - 1 + ... is very obviously not absolutely convergent!
12. 08 Oct '10 20:09
Originally posted by wolfgang59
I think I knew that!
ðŸ˜•
I know you know. And I know also that you know that I know that you know it.
13. wolfgang59
Mr. Wolf
08 Oct '10 20:27
Originally posted by FabianFnas
I know you know. And I know also that you know that I know that you know it.
I didnt know that!