Two More "New" Problem Solving Opportunities
I am passing on two more problem-solving opportunities sent to me by C.S. (Mount Vernon, WA)...
Question 1: For what value(s) of b is(are) the graphs of f(x)=bx and
Question 2: On what interval does the infinite power tower x^x^x^x... converge?
Please send me your insights and potential solutions, which I will both publish and pass on to C.S.
Hint: Sorry...none available...
Solution Commentary: Sorry...none available...until someone sends me a solution or two....But see commentary offered in "Archive" for August 30, 2009.
This commentary triggered a response from J.S. (Meridian, WA): "In regards to M.J.'s comments on the x^x^x^x... problem, I think that unless I'm missing something here I have a hard time with convergence with anything larger than one. It seems to me that convergence to 1 will happen with anything in the interval (0,1]. It also converges when x = -1, as I think that is the constant sequence -1, -1, -1,... . Furthermore, there is something interesting in the interval (-1,0). I haven't given any thought yet to the irrational values therein, but if we look at the rational values of the form (-1/n) where n is odd we might have something. I believe they may very well converge to the value -1. If n is even we have the little problem with complex numbers and although there may be convergence within the complex field, again I haven't pursued that possibility. Meanwhile, back at (-1/n) where n is odd, I offer this example: (-1/3)^(-1/3)^(-1/3)^(-1/3)... starts off with one over cube root of -3, ninth root -3, one over 27th root -3, and so on .... It seems to me that the even numbered terms converge to -1 from above and the odd numbered terms converge to -1 from the bottom. Granted, this does not constitute an interval, and I haven't really proved that they all converge, but I think it is worthwhile noticing. Also, I have not yet given any attention to ALL rational numbers in the interval. Maybe when other pressures ease, I may do some of the above."
When I responded "given your second to last sentence, it might take you a while to give 'attention to ALL rational numbers in the interval,'" J.S. counter-responded: "Honestly, I believe that all rational numbers with odd denominators in the interval [-1,0) converge. However, I think that those with even numerators converge to 1, not -1. As you pointed out, paying attention to ALL rational numbers is costly in matters of time. I have tried, and have reached the vicinity of "half-way to infinity". That's not so impressive when one considers that we are only talking aleph-null. Imagine the time I'll have to spend if I decide to pursue the behavior of the irrational numbers."