While surfing on YouTube, I stumbled into this video which gave me a new insight about the well-known series $$ \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7}+ \ldots $$ The idea shown there consists of counting (in a very clever way) how many points of the 2d integer lattice lie on a generic circumference of radius $\sqrt{r}, r \in \mathbb{N}$, centered at the origin. Then, we name $N(R)$ the number of points of this kind that lie inside the circumference of radius $\sqrt{R}$: as $R$ grows, we can think of $N(R)$ as a fairly good approximation of the area $\pi R$ of the circle, since each of the $N(R)$ points can be thought as the center of a square of area $1$. From the equality $\pi R = N(R)$ we get the series above.
I was fairly amazed by the way this result was obtained, and I started wondering: what if we used the hexagonal lattice (I mean $\mathbb{Z} \times \zeta_3\mathbb{Z}$, where $\zeta_3$ is a non-trivial third root of unity) instead of the integer lattice? Will I get another series to approximate $\pi$?
After some work, following the same kind of argument, I came to this formula: $$ \frac{\pi}{3} = \sqrt{3} \left (1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{5} + \frac{1}{7} - \frac{1}{8} + \ldots \right ) $$ or equivalently $$ \frac{\pi}{3} = \sum_{k=0}^\infty \frac{\sqrt{3}}{9k^2+9k+2} $$ I have some questions:
Can anyone give me a proof of this result which does not follow the argument I sketched above? I checked by doing some simple calculations (I also asked Wolfram) and the result seems to hold, but I'd like to be 100% sure...
The $\frac{\pi}{4}$ formula relates to the expansion of $\arctan(x)$, but I couldn't find any straightforward connection between the $\frac{\pi}{3}$ formula and $\arctan(x)$. I actually couldn't find any connection to the "pi facts" I know or I was able to find. Do anybody know anything that can explain "easily" what's happening here?
Thanks in advance!
