Concrete Math Sum 1

[rsoldo]

Calculate:

$$\sum_{k\in S}3^{-k}$$

$S$ is a set of all natural numbers that aren't divisible by either 2,3 or 5.

[maxal]

For any subset $T\subseteq\mathbb{Z}_{\geq 0}$, define

$$f(T) := \sum_{k\in T} 3^{-k}.$$

Computing $f(S)$ can be easily done by inclusion-exclusion:

$$f(S) = f(D_1) - f(D_2) - f(D_3) - f(D_5) + f(D_6) + f(D_{10}) + f(D_{15}) - f(D_{30}),$$

where $D_n$ is the subset of $\mathbb{Z}_{\geq 0}$ formed by multiples of $n$ (in particular, $D_1=\mathbb{Z}_{\geq 0}$). Namely, we have

$$f(D_n) = \frac{1}{1-3^{-n}},$$

from which the value of $f(S)$ follows instantly.