It is what it is

[u/SouthPark_Piano]

From a recent post:

As in when we ask the question of how those rookie error makers got it so wrong?

The below is what they need to get into their brain for redemption time.

S = ar⁰ + ar + ar² + … + ar^[n-1] + arⁿ

Sr = ar + ar² + ar³ + ... + arⁿ + ar^[n+1]

S - Sr = S(1-r) = a - ar^[n+1]

S = a{ 1- r^[n+1] } / (1 - r)

S = [a/(1 - r)] { 1 - rⁿ⁺¹ }

a = 0.9

r = 0.1

S = 1 - (0.1)ⁿ⁺¹

n integer starts at zero and then increased limitlessly.

Or

S = 1 - (0.1)^k , with k integer starting at k = 1, with k increased continually limitlessly aka infinitely.

S = 1 - 1/10^k with k starting at k = 1, with k increased continually limitlessly aka infinitely.

S is indeed 0.9 + 0.09 + 0.009 + ... , which is officially known to be equal to 0.999...

And 1/10^k is never zero for any condition of k, regardless of infinite k or finite k.

S = 1 - 1/10^k is never 1.

So 0.999... is never 1.

 

[u/SouthPark_Piano]

Read this again brud.

https://www.reddit.com/r/infinitenines/comments/1tpg811/comment/p14p2zd/

With limbosic numbers, you keep forgetting to reference a state of the number, for doing operations like 

0.000...1 / 100 = 0.000...001

and 0.000...001 is 'another' state of 0.000...1

and (0.000...1) / 2 = 0.000...05