Re: The truncated harmonic series diverges.

On 05.03.2025 18:18, efji wrote:

Le 05/03/2025 à 11:01, WM a écrit :

The harmonic series diverges. Kempner has shown in 1914 that all terms containing the digit 9 can be removed without changing the divergence.

Mistake. That means that the terms containing 9 diverge.

 ???
Kempner has shown in 1914 that the harmonic series CONVERGES if you omit all terms whose denominator expressed in base 10 contains the digit 9.

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.
Regards, WM