Does this geometricish sequence end/converge? ![enter image description here]( 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.

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.