Okay, maybe not billions. But by taking cosines repeatedly, one can find the solution of the equation with high precision in under a minute.

Step 1: Enter any number, for example 0, and press Enter.

Step 2: Enter `cos(Ans)`

and press Enter

Step 3: Keep pushing Enter. (Unfortunately, press-and-hold-to-repeat does not work on TI-83). This will repeatedly execute the command `cos(Ans)`

.

After a few iterations, the numbers begin to settle down:

and eventually stabilize at `0.7390851332`

**Explanation**: the graph of cosine meets the line at one point: this is a unique **fixed point** of the function .

Since the derivative at the fixed point is less than 1 in absolute value, the fixed point is **attracting**.

Now try the same with the equation .

This time, the numbers flat out refuse to converge:

**Explanation**: the graph of meets the line at seven point: thus, this function has seven fixed points.

And it so happens that at each of those fixed points. This makes them **repelling**. The sequence has nowhere to converge, because every candidate for the limit pushes it away. All that’s left to it is to jump chaotically around the interval . Here are the first 1024 terms, plotted with OpenOffice:

Clearly, the distribution of the sequence is not uniform. I divided the interval into subintervals of length and counted the number of terms falling into each.

What is going on here? Stay tuned.