Originally posted by wolfgang59Are there more even numbers than natural numbers?
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Of course!
I'm trying to point out the flaw in our opponents' argument in the "many balls" thread
Yet if I start with this sequence
{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"...
Originally posted by Palynkayes but did you know that there are more evens than odds?
Are there more even numbers than natural numbers?
Yet if I start with this sequence
{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"...
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
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Originally posted by wolfgang59inf - inf is not zero. It's not even defined. Therefore your calculation is flawed.
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]
If we define inf - inf, inf / inf, and such, as a number then we get funny results, as the one above.
Originally posted by ThomasterSo 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?
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
1 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!