The Riemann sphere at home

Sphere at home

Amongst friends.


The reality of life

Whenever you first meet someone, be it child, soulmate or stranger, we all know that, despite protests to the contrary, that it will end up only one way.

One of you will die before the other one.

It is a terrible thing to want to be the one that goes first or last. We just have to accept it, either way.

Remember you may now hold a unique set of memories and experiences that occurred after you first met. Try to keep them alive as long as possible.

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.