Q:

Does the dimension converge? 

I'm doing an exercise in a textbook and I need to calculate the effective dimension of a possibly non-Euclidean network lattice.  I have $r^d=2r^2+2r+1$ where $d$ is the dimension of the network and I need to count the number of nodes up to $r$ connections away from a given node.  I'm trying to figure out if $d$ converges as $r$ goes to infinity.  From my calculations it looks like $d$ might converge to $2,$ or something close to it.  

A:

For an upper bound, we have $d = \log_r(2r^2 + 2r + 1) \leq \log_r(5r^2) = 2\log_r(r) + \log_r(5) = 2 + \log_r(5) \to 2$ as $r \to \infty$. On the lower side, we have $d = \log_r(2r^2 + 2r + 1) \geq \log_r(2r^2) = 2 + \log_r(2) \to 2$ as $r \to \infty$. So yes, indeed, $d$ converges to $2$.

