Arithmetic in the height-parameter (sums, series)

[bo198214]

Well, the equality 1 + 2 + 4 + 8 + ... = -1, is very fragile if you want to take it as a serious argument. It doesnt surprise me that you dont get the result you expect.

Well ... "fragility"...., a bit informal.

Well I can also say *false*. Just wanted to pay respect that it miraculously works in several cases.

What I'm trying is to improve my general understanding of infinite series and especially that of divergent series (well in the context of iteration and tetration). Perhaps I went too far, in that I (unconsciously) moved nearer and nearer to a notion of "whenever a sum for a divergent series can be established (for instance by cesaro-sum), we can use that value in the instance of the series".

And this is the reason why I always avoided with alternate summability techniques in theoretical matter.
There is a well established theory about *usual* powerseries. They have a convergence radius, inside they always converge outside they always diverge.

If you now assign values also to the outside by some summation technique, its like you dont know what you do. It *can* have interesting results, but it may also fail, often it just lacks the theoretical base; though I admit it gives surprising results. But you never can use it as a serious argument e.g. in a proof; its fragile.

My understanding of assigning values outside the convergence radius is that of analytic continuation:
You have a powerseries e.g. 1+z+z^2+. It does not converge for |z|>1. This is due to a (n isolated) singularity at z=1. But of course it can be continued along a path from 0 to 2 avoiding 1. The powerseries is the development of 1/(1-z) at 0 and so we know the value at 2, it is -1.

In the case of non-isolated singularities, e.g. branch points, there may be different values (branches) that may be obtained by continuation. Different summation methods may then return different values.

In general a sum s_0 + s_1 + .... may be considered to be the continuation of a powerseries at some point to some point outside or on the boundary of the convergence disk.

More generally I think summation techniques just distort the region of convergence from a disk (powerseries) to some other shape (see e.g. the thread about the Mittag-Leffler star, this is also a resummation and has as its domain of convergence the whole plane without the rays going from singularaties outward (roughly spoken)).

But in my two examples we deal (with the limit of) functions of the partial sums and the condition "...if that converges..." must be extended to the additional condition "...if the sequence of function-evaluations of the partial sums converge..." or something like. (*2)

In your example you just obtain another inequality which you already get if not considered as iteration exponents.
\( 1+2+4+8+... = \infty \neq -1 \)
\( f^{\circ 1+2+4+\dots}(z) = a \neq f^{-1}(z) \)

(*1) I'm proudly linking to my own solution for the summation of 0!-1!+2!-3!... in eulerian-matrix chap 3.2

Seems matrix method is quite potent to sum everthing