Hi fellows -
just posted three msgs to sci.math. With the third msg it got much interesting, so I'll copy it to our forum, just without further comment.
(msg 1) "Another tetration-series for base sqrt(2)"
I didn't see this series before, so just for the record.
Seems to be the most simple form of a series for tetrates, which can be extracted from the diagonalization-approach.
The type of series is characteristic, but is base-dependent.
Here for base b = sqrt(2) :
where u=log(2), such that exp(u/exp(u)) = b and must be read as u_h = u^h
Note, that if h->inf, all except the first term of the series vanish so that 2 remains; if h=-1, that means u_h~ 1.442 the series seem to converge to zero and for h=-2 the series seems to diverge slowly to -infty. (which is obviously what we expect with tetration)
(msg 2)
Hmm, just found an argument for the last hypothese. Saying the series diverges for h=-2 to infinity reminds to the harmonic series. So let's see, whether a rescaling of the coefficients by u^2 tends to the harmonic series or a scalar multiple. To see the convergence better, the coefficients are also scaled by the reciprocals of natural numbers. Then I get the following series, where u_h2 means u^(h+2), such that for h=-2 all powers of u become 1
The coefficients seem to approximate 2.88539008178... = 2/u and we get indeed asymptotically the form of the harmonic series when h=-2.
Cute... This would be nice to be proven. But how? :-)
(msg 3)
Using 128 terms for the series, and 30 digits shown we get
Very nice...
Gottfried
just posted three msgs to sci.math. With the third msg it got much interesting, so I'll copy it to our forum, just without further comment.
(msg 1) "Another tetration-series for base sqrt(2)"
I didn't see this series before, so just for the record.
Seems to be the most simple form of a series for tetrates, which can be extracted from the diagonalization-approach.
The type of series is characteristic, but is base-dependent.
Here for base b = sqrt(2) :
Code:
b^^h = 2
- 0.632098661051 *u_h - 0.225634285681 *u_h^2 - 0.0854081730270*u_h^3
- 0.0335771160755*u_h^4 - 0.0135675339902*u_h^5 - 0.00559920683946*u_h^6
- 0.00235003288785*u_h^7 - 0.00100003647235*u_h^8 - 0.000430480708304*u_h^9
- 0.000187116458671*u_h^10 - 0.0000820114021745*u_h^11 - 0.0000362027647360*u_h^12
- 0.0000160807242165*u_h^13 - 0.00000718169500164*u_h^14 - 0.00000322271898338*u_h^15
- 0.00000145228984161*u_h^16 - 0.000000656926890186*u_h^17 - 0.000000298154140684*u_h^18
- 0.000000135730176453*u_h^19 - 0.0000000619577730720*u_h^20 - 0.0000000283522848887*u_h^21
- 0.0000000130033888474*u_h^22 - 0.00000000597608342584*u_h^23 - 0.00000000275165501559*u_h^24
- 0.00000000126917896344*u_h^25 - 0.000000000586333973928*u_h^26 - 0.000000000271274348008*u_h^27
- 1.25680399977 E-10*u_h^28 - 5.83015364711 E-11*u_h^29 - 2.70775077830 E-11*u_h^30
- 1.25898302877 E-11*u_h^31
- O(u_h^32)where u=log(2), such that exp(u/exp(u)) = b and must be read as u_h = u^h
Note, that if h->inf, all except the first term of the series vanish so that 2 remains; if h=-1, that means u_h~ 1.442 the series seem to converge to zero and for h=-2 the series seems to diverge slowly to -infty. (which is obviously what we expect with tetration)
(msg 2)
Hmm, just found an argument for the last hypothese. Saying the series diverges for h=-2 to infinity reminds to the harmonic series. So let's see, whether a rescaling of the coefficients by u^2 tends to the harmonic series or a scalar multiple. To see the convergence better, the coefficients are also scaled by the reciprocals of natural numbers. Then I get the following series, where u_h2 means u^(h+2), such that for h=-2 all powers of u become 1
Code:
b^^h =
2 +
-1.31563054605 / 1 * u_h2
-1.95493914978 / 2 * u_h2^2
-2.31029756506 / 3 * u_h2^3
-2.52057540946 / 4 * u_h2^4
-2.64981962459 / 5 * u_h2^5
...
-2.88539007305/ 50 * u_h2^50
-2.88539007576/ 51 * u_h2^51
-2.88539007762/ 52 * u_h2^52
-2.88539007891/ 53 * u_h2^53
-2.88539007980/ 54 * u_h2^54
-2.88539008041/ 55 * u_h2^55
-2.88539008083/ 56 * u_h2^56
-2.88539008113/ 57 * u_h2^57
-2.88539008133/ 58 * u_h2^58
-2.88539008147/ 59 * u_h2^59
-2.88539008156/ 60 * u_h2^60
-2.88539008163/ 61 * u_h2^61
-2.88539008168/ 62 * u_h2^62
-2.88539008171/ 63 * u_h2^63The coefficients seem to approximate 2.88539008178... = 2/u and we get indeed asymptotically the form of the harmonic series when h=-2.
Cute... This would be nice to be proven. But how? :-)
(msg 3)
Using 128 terms for the series, and 30 digits shown we get
Code:
-2.885390081777926814 49154212159
-2.885390081777926814 56186809057
-2.885390081777926814 61052960248
-2.885390081777926814 64420113651
-2.885390081777926814 66750068482
-2.885390081777926814 68362344014
-2.885390081777926814 69478019929
-2.8853900817779268147 0250066967
-2.8853900817779268147 0784331383
-2.88539008177792681471 154053478
-2.88539008177792681471 409912721
-2.88539008177792681471 586977937 at k=127
...
-----------------------|---------
-2.88539008177792681471 984936200 -2/log(2) assumed to be the lower boundVery nice...
Gottfried
Gottfried Helms, Kassel