Sum of Reciprocals of Divisors equals Abundancy Index

Theorem
Let $n$ be a positive integer.

Let $\map {\sigma_1} n$ denote the divisor sum function of $n$.

Then:
 * $\ds \sum_{d \mathop \divides n} \frac 1 d = \frac {\map {\sigma_1} n} n$

where $\dfrac {\map {\sigma_1} n} n$ is the abundancy index of $n$.