Just like I have always wondered whether there is a conceptual, not just etymological, relationship between the terms "linear algebra" and a "sigma algebra," (there isn't) I would like to know whether geometric series relate in any way to the notion of geometry.
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Yes, have a look at https://en.wikipedia.org/wiki/Square_root#Geometric_construction_of_the_square_root. – StubbornAtom Sep 09 '16 at 18:47
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If you know how to represent say,$\sqrt {21}$, geometrically, then it is the same as representing the geometric mean of $3$ and $7$. – StubbornAtom Sep 09 '16 at 18:49
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Do a google image search for "geometric mean" and then for "geometric series." – Qiaochu Yuan Sep 09 '16 at 19:07