## Franklin Pezzuti Dyer

### Beatty Series and Sums over Rationals

The exact value of the geometric series with ratio $x$ is well-known and easy to derive:

which converges for $|x|<1$. Recently I’ve been playing around with a similar type of series whose summand decreases almost geometrically, but with a twist. For example, consider the following series:

where $\alpha$ is some positive real constant, and the $\lfloor \cdot \rfloor$ denotes the floor or “greatest integer” function. When $\alpha\in \mathbb N$, this simply reduces to a geometric series, because $\lfloor \alpha n\rfloor = \alpha n$ when $\alpha$ is a natural number. Similarly, when $\alpha\in\mathbb Q$, it’s not quite a geometric series, but it can be reduced to a constant times a geometric series with a bit of casework. However, when $\alpha$ is irrational, this series is frustratingly difficult to calculate, and I haven’t been able to compute in closed-form a single non-trivial series of this form with $\alpha\notin \mathbb Q$ and $x\ne 0$.

Despite this difficulty, I’ve managed to come up with some interesting non-trivial identities relating series of this form to other series involving the expression $\lfloor \alpha n\rfloor$, which I’ll derive in this post.