So I just "learned" about the number Wau from Vi Hart's video. It's amusing, to be sure, but the actual "definition" she presents got me thinking.
We can formalize the construction in this way: set $x_0=\frac2{1+3}=\frac12$ and then $$x_{n+1} = \frac{2}{\frac12x_n+\frac32x_n} = \frac{2}{2x_n}=\frac{1}{x_n}.$$
Clearly, this sequence does not converge. However, if we assume it does converge to $x$, then $x=\frac1x$ and the only sensible solution to this is $x=1$.
I know that there are treatments of divergent series, and there are methods (for example, Cesàro summation) of assigning meaningful finite quantities to them, and I was wondering if the same is true for sequences. In particular, is there some more legitimate sense in which this sequence should be assigned the value $1$ than the simple "hope it converges" idea?
I tried to turn it into a series and then perform Cesàro summation; but as one would expect, this gives the arithmetic average of the distinct terms: $$x_n = 2 - \frac32\sum_{k=0}^n (-1)^k \xrightarrow{\text{Cesàro}} 2-\frac34 = \frac54$$
It may not be a correct intuition, but my idea is to perturb the iteration as follows: let $\epsilon > 0$ and define
$$ x_{0} = \tfrac{1}{2}, \qquad x_{n+1} = \frac{1}{x_{n}} + \epsilon. \tag{1}$$
Then it follows that $x_{n}$ converges to $\sqrt{1+\epsilon^{2}} + \epsilon$, which indeed converges to $1$ as $\epsilon \to 0^{+}$.
If we assume that this iteration procedure is given by some physics model, then it may make sense of thinking this perturbation. But this interpretation have some pitfall, in that (1) turns out to be very unstable if $\epsilon < 0$. Thus even if this viewpoint makes sense, we need a great care when we try to understand the meaning of the perturbation term $\epsilon$.