I'm looking for a simple proof or a reference to any proof that
For $j \in ℤ$, $0
My searches have turned up a few papers that appear to make use of this property, but none that prove it, refer to a source, or even give it a name.
EDIT:
I'm working with $k$ restricted to positive integer roots of rational numbers: $k=r^{1/m}=(a/b)^{1/m}$ with $r$ and $m$ chosen so that $m$ is the least positive integer for which $k^m \in ℚ$ (so when $k \in ℚ$, $m=1$).
The set of positive integer roots of rational numbers (set $) is not closed under addition (see the $\sqrt{2}-1$ case in comments) Given $l$, $h$ members of $, $\left(l\pm h\right) = l \left(1\pm{h\over l}\right) = l \left(1+k\right) \notin $ occurs when $\left(1+k\right) \notin $.
The sum before the edit is a rearrangement of the binomial expansion of $\left(1+k\right)^n$ where a different choice of $n$ changes the values of $d_j$, and excluding the rational $j=0$ term. If the sum is irrational, then $\left(1+k\right)^n \notin ℚ$ for any $n$, $\left(1+k\right) \notin $, and $\left(l\pm h\right) \notin $ occurs when $\pm {h\over l} = k \notin ℚ$.