The harmonic series diverges. Kempner has shown in 1914 that when all
terms containing the digit 9 are removed, the series converges. Here is
a simple derivation:
https://www.hs-augsburg.de/~mueckenh/HI/ p. 15.
That means that the terms containing 9 diverge. Same is true when all
terms containing 8 are removed. That means all terms containing 8 and 9 simultaneously diverge.
We can continue and remove all terms containing 1, 2, 3, 4, 5, 6, 7, 8,
9 in the denominator without changing this. That means that only the
terms containing all these digits together constitute the diverging series.
But that's not the end! We can remove any number, like 2025, and the
remaining series will converge. For proof use base 2026. This extends to
every definable number. Therefore the diverging part of the harmonic
series is constituted only by terms containing a digit sequence of all definable numbers.
Note that here not only the first terms are cut off but that many
following terms are excluded from the diverging remainder.
This is a proof of the huge set of undefinable or dark numbers.
Regards, WM
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