geometric series diagram query.

Question

I was going through this article on geometric series on Wikipedia and found this diagrammatic representation of an infinite geometric series with a said common factor of (1/2) but shouldn't the common factor be (1/4) based on the diagram? link to the diagram: https://commons.wikimedia.org/wiki/File:GeometricSquares.svg#/media/File:GeometricSquares.svg

Answer 1

Yes, but then that's what the author says: the first term of the series is $\frac12\times\frac12=\frac14$, the second one is $\frac14\times\frac14=\frac1{16}$, and so on. And its sum is$$\frac{1/4}{1-1/4}=\frac{1/4}{3/4}=\frac13.$$

Answer 2

The series in the introduction of the article geometric series and the sequence depicted in your diagram are not the same. The one in the introduction is $\sum_{n = 1}^{\infty} \frac{1}{2^n}$ while the one in the diagram is $\sum_{n = 1}^{\infty} \frac{1}{4^n}$.

So, yes, the common factor is $\frac{1}{4}$ as you say.