Exponential Explorations – Part II
(Part I is here.)

I want to thank everyone who gave Part I a try and encourage everyone who didn’t to at least follow along. There will be bits of this that everyone will be able to do, though other bits will be quite challenging.

To review the solution of Part I, the insight lies in realizing that if the chain of exponents is infinite, then adding another x at the bottom makes no difference. So
f(x) = x ^ f(x)
If we want f(x) to be equal to 2, we just plug 2 in for f(x).
2 = x ^ 2, so x = sqrt(2).

Let’s demonstrate that numerically and play with some different values of x. This is where an Excel spreadsheet comes in handy. Open a new one. (You may be able to do this with a calculator, but it’ll be a pain.) In cell A2 (we’re leaving A1 blank for now) type

=sqrt(2)

and hit enter. The cell should display a number around 1.4142. Widen the A column by dragging the thin line between the header of column A and column B so that you can see as many of the digits of the square root of 2 as Excel can calculate. (You can also double click on that line. That automatically sizes a column to the width of its fattest cell.) Now in cell A3 type

=$a$2^a2

and hit enter. Quick Excel lesson: the dollar signs make the first cell reference absolute, meaning “no matter where this formula gets copied, always point to cell A2.” The second cell reference without the dollar signs is relative, meaning “wherever this formula gets copied, always point to the cell one above the cell that the formula is in.”

Cell A3 should have a number about 1.63 if you’ve done everything right. Now click back on cell A3 to select it. The right lower corner of the selected cell is a square black drag box that I circled below for you.



That drag box lets you easily copy from one cell to others. Grab the drag box with your pointer and drag it straight down the A column so that you cover about 100 or two hundred cells. What you have now is a whole bunch of cells that each calculate “the value in A2 taken to the power of the cell immediately above me”. So if we put x in A2, the rest of the column calculates progressively close approximations of f(x).

You should be able to scroll down and see that the numbers eventually converge to 2 and remain 2 ever after, which is just the result we were hoping for.

With me so far? Good.

Now what if in Part I asked you to find x such that f(x) = 3? Using the same solution we would come up with x = the cube root of 3, or 3^(1/3). Let’s plug that in. In cell A2 type

=3^(1/3)

and press enter. A2 becomes about 1.442 and the entire column recalculates, and we scroll down eagerly anticipating the numbers in the column to settle down to 3. But they don’t. They settle to about 2.478.

What’s going on here?

Let’s see if we can get a higher answer by just plugging in bigger numbers. If we type 2 in to A2, the entire function blows up after just three iterations. That makes sense. f(2) just gets bigger without limit and doesn’t converge. Try 1.5 in A2. Same thing. It blows up, but it takes a little longer. But the square root of two and the cube root of three which are over 1.4 but less than 1.5 both converge.

Here’s the challenge for this part, and I admit, it’s hard. Find the upper limit of convergence of f(x) exactly. That is, find the highest x for which f(x) converges to some finite number. You can play with the spreadsheet to give you some ideas, but ultimately to get an exact (not a numerical) answer, you’ll need to put the spreadsheet away and do some math. By exactly, I mean give the answer as a mathematical expression, not as a number. For example the exact answer of Part I was “the square root of 2” whereas 1.41 (or any number of digits) would not have been exact.

Feel free to use the comments to collaborate with others and share ideas or partial results. If there’s no progress, every day or two I’ll give you a step towards the solution. - served up by Doctor Bean @ 10:29 PM