Re: Replacement of Cardinality

Am 27.08.2024 um 22:14 schrieb Chris M. Thomasson:

On 8/27/2024 1:01 PM, Moebius wrote:

Am 27.08.2024 um 21:34 schrieb Chris M. Thomasson:

On 8/27/2024 12:20 PM, WM wrote:

Le 25/08/2024 à 23:18, Richard Damon a écrit :

>
But the sum [...] adding up the terms of
>
1/n - 1/(n+1)
>
never gets to 1,

>
1/1 - 1/2 = .5
1/2 - 1/3 = .1(6)
1/3 - 1/4 = .08(3)

>
Don't forget to sum up the terms:
>
     (1/1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) = ?

 3/4
 Yup. Never equals one... However it "tends" to it...

Right. Actually, lim_(n->oo) SUM_(k=1..n) 1/k - 1/(k+1) = 1.
Hence we write "SUM_(k=1..oo) 1/k - 1/(k+1) = 1" and speak of an "infinite series" with sum 1 (thouh we do not actually "sum up" infinitely many terms).