Arithmetic in the height-parameter (sums, series)

[Gottfried]

Hi -

using the notation \( x_1 = \exp_b^{^{oh_1}}(x_0) \) and \( x_2 = \exp_b^{^{oh_2}}(x_1) \) we do arithmetic in the height (or "iteration") parameter like \( x_2 = \exp_b^{^{oh_1+h_2}}(x_0) \)

What about infinite series instead of a sum?
If we have a sufficient method for continuous tetration, then, for instance we should get
\( x_2 = \exp_b^{^{o1/2+1/4+1/8+...}}(x_0) = \exp_b^{^{o1}}(x_0) = b^{x_0} \)

For "nice" bases 1<= b <=ß (= exp(exp(-1)) ) we should get even more; I don't see any reason, why the series in the height-parameter should not be allowed to diverge.

So, with base b=sqrt(2) the following expression

\( y = \exp_b^{^{o1 + 2 + 4+ 8 +...}}(x_0) \)

seems to make sense to me because the partial evaluations converge to y=2 , for instance if we choose x_0 = 1.

On the other hand, the analytical continuation for the geometric series with constant quotient q \( g(q) = 1+q+q^2+ ... \) at q=2 gives

\( g(2) = 1/(1-2) = -1 \)

But -substitued this into the height-parameter- then we should also have

\( y = 2 = \exp_b^{^{o -1 }}(x_0) = \log_b(x_0) = \log_b(1) = 0 \)

where we see a contradiction.
So it seems to me that tetration can restrict the validity/range-of-usability of analytic continuation (if we do not have another explanation/notion/way-of-handling of such effects).

---

In a second view, this extends to other functions similarly. We have another inequality if a divergent series is used in the exponent of the power-function, for instance, to base e:

\( y = e^{^{-1 -2 -4 -8 -16 - \dots }} = \frac1{e^1}*\frac1{e^2}*\frac1{e^4} * \dots = 0 \neq e^1 \)


Just another plot of meditations... <phew>

Gottfried

[bo198214]

For "nice" bases 1<= b <=ß (= exp(exp(-1)) ) we should get even more; I don't see any reason, why the series in the height-parameter should not be allowed to diverge.

So, with base b=sqrt(2) the following expression

\( y = \exp_b^{^{o1 + 2 + 4+ 8 +...}}(x_0) \)

seems to make sense to me because the partial evaluations converge to y=2 , for instance if we choose x_0 = 1.

On the other hand, the analytical continuation for the geometric series with constant quotient q \( g(q) = 1+q+q^2+ ... \) at q=2 gives

\( g(2) = 1/(1-2) = -1 \)

But -substitued this into the height-parameter- then we should also have

\( y = 2 = \exp_b^{^{o -1 }}(x_0) = \log_b(x_0) = \log_b(1) = 0 \)

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.

[Gottfried]

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.

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".
That's obviously a false mental notion - although it is suggestive and is working in many cases. (*1)

Perhaps a step towards a better understanding is the following consideration.

Assume the sequence of partial sums of some divergent series a0+a1+a2+... or even better, using some limit notation like
f(x) = a0+a1 x + a2 x^2 + ... for x->1(-)
Then for some divergent series we take the mean of the sequence of partial sums, if that converges.

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)

Hmm. I think to proceed here I should do some examples, where the function-evaluations converge and some, where they don't ...

Gottfried


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

[update]
(*2) well, again in a second thought: I was actually thinking about that, when I posted this whole problem: we see, that the tretration-function even provides this convergence of sequence of function-evaluations. And still this is not enough to guarantee the usability of the "value" of the divergent series...
[/update]

[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

[Gottfried]

(*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

- unfortunately not... That was subject to a very recent post to sci.math, where I asked, whether there is a known limit for the summation-power of triangular matrices for matrix-based transform & summation. I didn't find a possibility for a more powerful matrix and suppose, that such a limit might already be known.

Well, back to the main subject. Surely, the series 1+2+4+8+... is a "fragile" or let's say, basing arguments on it is fragile. I chose that only because of its simple occurence. Generally I think, it should be allowed to formulate the iteration parameter as a series, and also as a powerseries in x, for instance

\( f(x) = \exp_b^{^o ^{1+x+x^2+x^3+...}} (a) \)

and then discuss the limiting behaviour, when abs(x)->1(-) . One of the reasons, that the summation 1-1+1-1+... made it into serious math, and was accepted to be identified with the value 1/2 was the discussion of the form of g(x) = 1+x+x^2+... for x=-1, the form of the geometric series, its translation into the closed form 1/(1-x) and finally the method of analytical continuation.
But can we identify the above f(-1) with \( f(-1) = \exp_b^{^o ^{0.5}}(a) \) using the rationale of evaluation using the partial sums?
Would be interesting, whether this makes sense, anyway ...

So do we deny the validity of divergent summation in the height-parameter completely? Then we should do it also explicitely, for instance also in a remark in wikipedia or other online-resources.
But I think, that were a step too early.
[Update] a) What would we do in cases, where the height-parameter is expressed as zeta-series. Zeta-regularization is a well established procedere. Does it produce contradictions if inserted in the height-parameter in tetration (or other iterated functions) ?[/update]
b) We should in general look, whether there are possibilities, where divergence/summation keeps a sensical result, or if the contrary occurs, and we cannot find any such meaningful result, then we should try to explain, why the consideration near the limit can*not* be extended beyond (or some wording like "analytical continuation makes no sense here")

Hmm, perhaps I'd look for opinions in sci.math, too...

Gottfried

[bo198214]

Well, back to the main subject. Surely, the series 1+2+4+8+... is a "fragile" or let's say, basing arguments on it is fragile. I chose that only because of its simple occurence. Generally I think, it should be allowed to formulate the iteration parameter as a series, and also as a powerseries in x, for instance

\( f(x) = \exp_b^{^o ^{1+x+x^2+x^3+...}} (a) \)

and then discuss the limiting behaviour, when abs(x)->1(-) .

Of course it is allowed. In this context a series would be considered as a sequence, the sequence of the partial sums; and a good continuous iteration of a function should be continuous in the exponent, i.e.
\( \lim_{n\to\infty} f^{\circ s_n}(x) = f^{\circ \lim_{n\to\infty} s_n}(x) \).

But the partial sums of 1+2+4+8+... as well as the partial sums of 1-1+1-1... diverge.
At most one could say that the partial sums of 1+2+4+8+... converge to oo (e.g. in the topology of the complex sphere convergance to infinity has a well defined meaning.)

So by continuity of the iteration exponent, one would expect
\( f^{\circ 1+2+4+8+\dots} (x) = f^{\circ \infty}(x) \)
and one would expect
\( f^{\circ 1-1+1-1}(x) \) to diverge.
Which is indeed the case.

If you choose the partial sums differently, e.g. by calculating the mean or so then perhaps 1-1+1-1... converges to 1/2 and so this would happen if raised to the iteration exponent (in which case you however would lose the integer exponents).

One of the reasons, that the summation 1-1+1-1+... made it into serious math, and was accepted to be identified with the value 1/2.

This is not true. The partial sums of 1-1+1-1... do not converge, the series is divergent. Whether Euler wrote something different in his time is another thing. Calculus wasnt so precise at this time, and in a *certain way* it makes sense to assign the value 1/2, but in the default meaning of what every freshman learns in the first analysis course about series it is provably wrong, false, not true, incorrect, ...

[Update] a) What would we do in cases, where the height-parameter is expressed as zeta-series. Zeta-regularization is a well established procedere. Does it produce contradictions if inserted in the height-parameter in tetration (or other iterated functions) ?[/update]
b) We should in general look, whether there are possibilities, where divergence/summation keeps a sensical result, or if the contrary occurs, and we cannot find any such meaningful result, then we should try to explain, why the consideration near the limit can*not* be extended beyond (or some wording like "analytical continuation makes no sense here")

There is nothing special about taking limits in the iteration exponent, if you take any continuous function h, the same thing occurs:
h(1+2+4+...)=h(oo)
h(1-1+1-1+...) diverges

but sequences of integer numbers never have non-integer numbers as limit.

[Gottfried]

Hmm, some points come out clearer now.
What I want is not sort of legimitation for this or that but of understanding. I'm ok, if it comes out that we must dismiss any sort of divergent summation in the iteration parameter of functions. What I want is: is this necessary? Are there exceptions?
If this is a general property, I also want, that we add some remark to our interpretation of tetration: "no divergent series in iteration parameter" - be it in open-text collections like wikipedia or in journals.
Since it seems, that is a more general property, I'd like to see this also in articles about summation of divergent series: "that concept is limited to <...something...>" and "not meaningful for <...example:iteration of functions...> " (I need not Euler to assign a fairly general availability of divergent summation - K.Knopp and G.H.Hardy have even dedicated monographies (or monographic-like chapters) to that concept - without mentioning circumstances, where it is *generally* not applicable)

But well, let's see. I think I'll do some more examples first, to improve my own understanding. I'll reply to this all later again.

Gottfried

[bo198214]

If this is a general property, I also want, that we add some remark to our interpretation of tetration: "no divergent series in iteration parameter" - be it in open-text collections like wikipedia or in journals.

No, thats not the way it goes. 1+2+4+... = -1 is false in the normal sense of the limit; if it is despite useful in certain circumstances this needs to be mentioned and not the opposite that it is not useful here.

So if you find a certain way to use summation methods in the iteration exponent and getting interesting results - great, but nobody would expect that it works by default.

(I need not Euler to assign a fairly general availability of divergent summation - K.Knopp and G.H.Hardy have even dedicated monographies (or monographic-like chapters) to that concept - without mentioning circumstances, where it is *generally* not applicable)

I didnt read such a monograph, but I can not imagine (from what I read by Hardy or Knopp) that they write something that is formally false, like 1+2+4+...=-1.

One term from analysis is for example "absolutely convergent", where you can reorder the summands in any way without affecting the limit. But non-absolutely convergent series is still a convergent series but the limit (of the partial sums) may depend on reordering. Here we not even work with divergent series, and we use no complicated summation method but just reorder the terms - but despite they give different limits. Taking this further you could prove that every number is equal to any other number.