Riemannian geometry. Sphere v Plane


Does the Harmonic series converge or diverge in the ‘real’ world?
“In mathematics, the harmonic series is the divergent infinite series:

It is well known that the Harmonics series diverges in the pure world (see above).

Does it also converge to a finite value in the ‘real’ world? I think so. Anyone want to disagree?

Using ≜ to indicate an origin then:-

The pure number range for the Integer series is

± {∞:n/1:1:≜}

the pure number range for the Harmonic series is

± {∞:1/n:1:≜}

These 2 can be combined to produce all floating point numbers.

± {∞:f/1:1.0:1/f}, ± {∞:f/1:{≜}:1/f}

The number ranges that are practical (real) to work with, measured in SI units, are only a sub-set of the above though

± {limit:n/1:1:≜}


± {limit:1/n:1:≜}

Therefore, by replacing the usual work in n/1, and by keeping track of the scales and meanings that have been applied, one can work simpler in 1/n to whatever resolution and precision the limit to limit range implies.

This also goes for all constants chosen from ± {∞:n/1:1:≜} such as π, √2, e and i.

≜ is often called/replaced by 0.
One lays over the other to provide an origin to the number series if including 0 is all.
It is much simpler, faster and more detailed to work in 1/n, even for y ≜ f(x), with limits properly applied that is.
This removes 0 from consideration as it is replaced by a range scale change.
0 and ∞ can then be correctly reported as an error as they cannot be reproduced in ± {limit:1/n:1:≜}

A pure ± {0:1} 1/SI units world.

This then means that the Harmonic series converges to a finite value in the ‘real’ world determined by the measurement limits/length to the number of digits used to record the data.