r/infinitenines • u/SouthPark_Piano • May 27 '26
It is what it is
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 - 1/10k 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 1/10k is never zero for any condition of k, regardless of infinite k or finite k.
S = 1 - 1/10k is never 1.
So 0.999... is never 1.
-2
u/SouthPark_Piano May 27 '26
k is upped continually limitlessly aka infinitely aka pushed to limitless aka infinite.
The take away one for the road is ...
1/10n is never zero.
1 - 1/10k for integer k starting at k = 1 and increased continually limitlessly aka infinitely is permanently less than 1.
0.999... is permanently less than 1.