Does this geometricish sequence end/converge?

Question

Suppose a defined a kind of recursive/seqeunce definition. The first term is labeled as one. The second term is labeled as figure 2. The third term is labeled as figure 3. And we can do this until infinity. We're basically superimposing squares and circles and each subsequent superimposition fits in the previous term of the sequence. Does this recursively defined sequence converge and if it does, can we determine what shape the sequence will end on? Will my last shape in the sequence be a square or circle? We can only have squares and circles in the sequence even though I'm drawing it not to scale.

Answer 1

Assuming the $n$th term is just the innermost circle or square at each step, the sequence converges to the point at the center of the original circle but the sequence does not end. No matter how small the shapes get, you can always fit more squares and circles inside them.

There is no "last" term of the sequence, so it is meaningless to ask whether the last term is a square or a circle.

Answer 2

Let $r_n$ be the radius of the $n$th circle, and let the first circle have radius $r_1$. Then, using elementary geometry, you can show that the side length of the square inscribed in the circle with radius $r_i$ is given by $r_i\sqrt{2}$, and then that the radius of the circle inscribed in that square is given by $\frac{r_i\sqrt{2}}{2}$. Thus the radii of the circles form a geometric sequence with common ration $\frac{\sqrt2}{2}$, since for each $i$, $$r_{i+1}=\frac{\sqrt2}{2}r_i$$

EDIT: In the comments, you ask if "the last term when the picture reaches the center the origin point" will "end on" a circle or square.

The answer: it will never reach the origin point. No matter how small you get, the radius of the circle will never be $0$. That is the nature of the geometric sequence - it never reaches $0$, because you never multiply it by $0$.

Answer 3

Questions about convergence are complicated, since there are many forms of convergence, that rely on how you measure distance. This is because convergence talks about how close two objects are to each other, but objects can be similar in many ways, and convergence change depends on what property you are looking at. In the above example, if we consider two objects in the sequence close if the difference in there area is small, then your sequence converges to an object with zero area, i.e. a point. However, if you are looking at what type of curve you get at the end of a process, a square or a circle, you need to specify what your "distance between curves" is.