If [math] \sum\limits_{n=1}^{|\mathbb{N}|} 2^{-n} = 1 [/math],

>>8168992
1

Can you define what it means to sum a series over uncountably many terms?

But it doesn't equal 1. It's asymptotic.

>>8169027
You add unaccountably many terms together.

>>8169036
Addition is a linear operation that requires enumeration. So you can't.

>>8168992
I am not used to your notation

why |N|? and on the left N starts at 1 and on the right at 0?

so I will interpret it my own way

the one on the left is obvious

the one on the right is [math]2^{-2}+2^{-4}+...[/math]

this equals [math]\sum_{i=1}^{\infty}2^{-2i}[/math]

very easy to solve... 0.25 is half of 0.5 and so on. you can immediately recognize the proportions, that means

[math]\sum_{i=1}^{\infty}2^{-2i}=\frac{1}{3}[/math]

>>8169046

>on the left N starts at 1 and on the right at 0?

N doesn't start anywhere. n starts at one in both cases.

>>8169046
|N| is the number of natural number = infinity

>>8169046
ooops did I misread the OP?

it's 2^N, not 2N

so you mean [math]2^{-1}+2^{-2}+2^{-4}+2^{-8}+... = \sum_{i=1}^{\infty}2^{-2^{i}}[/math]

sooo yeah that will be a bit smaller than 1/3

I dont know how to solve this precisely but wolframalpha says 0.316422

if N includes 0 then the answer will obviously be 0.816422

>>8169051
well the equasion is wrong but you get the point

I cant remember the last time I did math so I am pretty shit at it