# Iterated exponentiation

• April 23rd, 2011, 06:38 AM
m84uily
Iterated exponentiation
Is there any way to find out what the following becomes?

• April 23rd, 2011, 01:35 PM
Arcane_Mathematician
Re: Iterated exponentiation
Quote:

Originally Posted by m84uily
Is there any way to find out what the following becomes?

By this, do you mean, for n=6, or
• April 23rd, 2011, 05:07 PM
m84uily
Re: Iterated exponentiation
Quote:

Originally Posted by Arcane_Mathematician
By this, do you mean, for n=6, or

Well the first one, I think. But just to clarify:

Rather than:

• April 23rd, 2011, 07:28 PM
MagiMaster
It should just approach 1. The nth root of anything less than 1 is closer to 1.
• April 23rd, 2011, 09:35 PM
m84uily
Quote:

Originally Posted by MagiMaster
It should just approach 1. The nth root of anything less than 1 is closer to 1.

I don't think this is the case, it seems that the number lies somewhere between .65 and .7

However if it was computed left to right and became then I would expect it to approach 1.
• April 24th, 2011, 05:16 PM
MagiMaster
Oh, right. I was reading it wrong. It might not converge. It looks like the evens and odds give two different answers. I don't think there's an easy way to calculate a closed form for either though. Power towers are typically complicated.
• April 24th, 2011, 05:40 PM
c186282
I agree with m84uily the result is and it does approach 1 very quickly

Here are the first few values:

0.79370052598409973738
0.94387431268169349664
0.98851402035289613536
0.99807644357562873885
0.99972497942011085466
0.99996561829044151083
0.99999617975167238799
0.99999961797451049383
0.99999996527040401417
0.99999999710586695511
0.9999999997773743809
0.9999999999840981701
0.9999999999989398780
0.9999999999999337424
0.9999999999999961025
0.9999999999999997835
0.9999999999999999886
0.9999999999999999994
• April 24th, 2011, 05:49 PM
MagiMaster
You're reading it the way I did at first, but that's not actually what he's asking.

The series would be:
(1/2) = 0.5
(1/2)^(1/3) = 0.7937
(1/2)^((1/3)^(1/4)) = 0.5906
...

The position of the parenthesis matters quite a bit when dealing with power towers.
• April 24th, 2011, 07:45 PM
c186282
Yes your right, MagiMaster it is more involved. However it does settle down kind of

http://dl.dropbox.com/u/161256/power.png

I do not know how to get at the limit analyticity but here are the computer values.

2. 0.5
3. 0.793701
4. 0.590564
5. 0.739735
6. 0.622665
7. 0.717776
8. 0.637556
9. 0.706797
10. 0.645562
11. 0.700619
12. 0.650267
13. 0.696884
14. 0.653189
15. 0.694526
16. 0.655061
17. 0.693002
18. 0.656277
19. 0.692011
20. 0.657069
21. 0.69137
22. 0.657578
23. 0.69096
24. 0.6579
25. 0.690704
26. 0.658098
27. 0.690549
28. 0.658217
29. 0.690458
30. 0.658285
31. 0.690406
32. 0.658324
33. 0.690377
34. 0.658345
35. 0.690362
36. 0.658355
37. 0.690354
38. 0.658361
39. 0.69035
40. 0.658363
41. 0.690349
42. 0.658365
43. 0.690348
44. 0.658365
45. 0.690347
46. 0.658365
47. 0.690347
48. 0.658366
49. 0.690347
50. 0.658366
51. 0.690347
52. 0.658366
53. 0.690347
54. 0.658366
55. 0.690347
56. 0.658366
57. 0.690347
58. 0.658366
59. 0.690347
60. 0.658366
61. 0.690347
62. 0.658366
63. 0.690347
64. 0.658366
65. 0.690347
66. 0.658366
67. 0.690347
68. 0.658366
69. 0.690347
70. 0.658366
71. 0.690347
72. 0.658366
73. 0.690347
74. 0.658366
75. 0.690347
76. 0.658366
77. 0.690347
78. 0.658366
79. 0.690347
80. 0.658366
• January 4th, 2012, 07:58 AM
amit28it
what this list is for ?