I have another strange, but again very basic result for the alternating series of powertowers of increasing height (I call it Tetra-series, see also my first conjecture at alternating tetra-series )

Assume a base "b", and then the alternating series

and for a single term, with h for the integer height (which may also be negative)

which -if h is negative- actually means (where lb(x) = log(x)/log(b) )

My first result was, that these series have "small" values and can be summed even if b>e^(1/e) (which is not possible with conventional summation methods). For the usual convergent case e^(-e)<b<e^(1/e) the results can be checked by Euler-summation and they agree perfectly with the results obtained by my matrix-method.(see image below)

Now if I extend the series Sb(x) to the left, using lb(x) = log(x)/log(b) for log(x) to base b, then define

This may be computed by the analoguous formula above to that for Mb from the inverse of Bb:

I get for the sum of both by my matrix-method

or, and this looks even more strange (but even more basic)

x cannot assume the value 1, 0 or any integral height of the powertower b^b^b... since at a certain position we have then a term lb(0), which introduces a singularity.

Using the Tb()-notation for shortness, then the result is

and is a very interesting one for any tetration-dedicated...

Gottfried

An older plot; I used AS(s) with x=1,s=b for Sb(x) there.

(a bigger one AS

Assume a base "b", and then the alternating series

Code:

`. `

Sb(x) = x - b^x + b^b^x - b^b^b^x +... - ...

and for a single term, with h for the integer height (which may also be negative)

Code:

`.`

Tb(x,h) = b^b^b^...^x \\ b occurs h-times

which -if h is negative- actually means (where lb(x) = log(x)/log(b) )

Code:

`.`

Tb(x,-h) = lb(lb(...(lb(x))...) \\ lb occurs h-times

-------------------------------------------------------

My first result was, that these series have "small" values and can be summed even if b>e^(1/e) (which is not possible with conventional summation methods). For the usual convergent case e^(-e)<b<e^(1/e) the results can be checked by Euler-summation and they agree perfectly with the results obtained by my matrix-method.(see image below)

Code:

`matrix-notation`

Sb(x) = (V(x)~ * (I - Bb + Bb^2 - Bb^3 + ... - ...) )[,1]

= (V(x)~ * (I + Bb)^-1 ) [,1]

= V(x)~ * Mb[,1] \\ (at least) for all b>1

= sum r=0..inf x^r * mb[r,1]

serial notation

= sum h=0..inf (-1)^h* Tb(x,h) \\ only possible for e^(-e) < b < e^(1/e)

\\ Euler-summation required

-------------------------------------------------------

Now if I extend the series Sb(x) to the left, using lb(x) = log(x)/log(b) for log(x) to base b, then define

Code:

`.`

Rb(x) = x - lb(x) + lb(lb(x) - lb(lb(lb(x))) +... - ...

This may be computed by the analoguous formula above to that for Mb from the inverse of Bb:

Code:

`.`

Lb = (I + Bb^-1)^-1

I get for the sum of both by my matrix-method

Code:

`.`

Sb(x) + Rb(x) = V(x)~ *Mb[,1] + V(x)~ * Lb[,1]

= V(x)~ * (Mb + Lb)[,1]

= V(x)~ * I [,1]

= V(x)~ * [0,1,0,0,...]~

= x

Sb(x) + Rb(x) = x

Code:

`.`

0 = ... lb(lb(x)) - lb(x) + x - b^x + b^b^x - ... + ...

x cannot assume the value 1, 0 or any integral height of the powertower b^b^b... since at a certain position we have then a term lb(0), which introduces a singularity.

Using the Tb()-notation for shortness, then the result is

and is a very interesting one for any tetration-dedicated...

Gottfried

-------------------------------------------------------

An older plot; I used AS(s) with x=1,s=b for Sb(x) there.

(a bigger one AS

Gottfried Helms, Kassel