Each of the purple squares has 1/4 of the area of the next larger square (1/2×1/2 = 1/4, 1/4×1/4 = 1/16, etc.). The sum of the areas of the purple squares is one third of the area of the large square.
https://en.wikipedia.org/wiki/Geometric_series#/media/File:GeometricSquares.svg
I'm wondering whether the geometric series shown is Cauchy. It eventually reaches a limit that is on the upper right hand corner. Converges to the black dot on the upper right hand corner.
Similarly, if the sequence shown below continues and approaches the origin, the center of the picture, I think the picture below is "pell series"?, does the "pell series" converge because it is Cauchy and reaches the center of the graph eventuall?
hint
For $n,p>0$, Let $$S_n=1/4+1/4^2+...1/4^n . $$
$$S_{n+p}-S_n=$$ $$1/4^{n+1}\Bigl(1+1/4+...1/4^{p-1}\Bigr) $$
$$=1/4^{n+1}\frac {1-1/4^p}{1-1/4} $$
$$\le \frac {1}{3.4^n}. $$
thus
$$\forall \epsilon>0 \;\; \exists N\in\mathbb N \;\;: \forall n>N \;\;\forall p>0$$ $$|S_{n+p}-S_n|\le \frac {1}{3.4^n}<\epsilon $$
because $\lim_{+\infty}\frac {1}{3.4^n}=0$. $(S_n)$ is Cauchy...