I noticed that too. I think it was probably just a typo. I mean, she really meant to write ((0))1 = 0 and ((0))2 = 1 but apparently interchanged them. (Mathematicians make those kind of mistakes too... ^^)Akonyl wrote: not to mention, you define ((x))3 as x^x^x, which means that ((x))2 would be x^x, which means that ((x))1 is just x, in which case, how is ((0))1 == 1? I could see maybe if you meant ((0))0 though, or if you improperly defined ((x))3 as x^x^x.
While 0.1 seems to converge, 1.1 does not actually go to +inf, it actually converges and this becomes apparent once you reach the 10th iteration. The 7th to 11th values are: 1.111781994, 1.111782009, 1.11782011, 1.11782011, 1.11782011(They aren't the same, the calculator just can't display further decimal places anymore.)Akonyl wrote: also, I'm not sure what you mean by "Try 1.1, or 0.1", because those seem to fit the assumption of "1.1 goes to +inf, and 0.1 converges" that you said they break, so
From what I have noticed, the closer the value of x to 0, the more the values seem to oscillate uniformly.
At x=0.01: 0.01, 0.95, 0.01, 0.94, 0.01, 0.94 (and it seems to diverge).
Actually, 0.1 seems to oscillate at first but it dampens since it is convergent. (The first 10 values are (in two decimal places): 0.10, 0.79, 0.16, 0.69, 0.20, 0.63, 0.24, 0.58, 0.26, 0.55)
The closer the value of x to 1, the less it oscillates and it dampens faster (or the less the differences between the odd and even iterations. For example, at x=0.5: 0.50, 0.71, 0.61, 0.65, 0.64, 0.64.
At higher values, the more apparent that it diverges.
Also, according to the wiki article that you posted,
Wikipedia wrote:In general, the infinite power tower, defined as the limit of nx as n goes to infinity, converges for e−e ≤ x ≤ e1/e, roughly the interval from 0.066 to 1.44, a result shown by Leonhard Euler.
That sums up my observations. But I'm still looking forward to what Giogio has to say, and yes, the proof.
