TITLE: Is sum of roots of 2 always irrational?
QUESTION [9 upvotes]: Define the sequence $r(k)= \sum_{n=2}^k 2^{\frac{1}{n}}$

Is $r(k)$ irrational for every natural $ k\geq 2$?

REPLY [4 votes]: Yes, because the field trace of $ \mathbf Q(2^{1/2}, 2^{1/3}, \ldots, 2^{1/n})/\mathbf Q $ kills this number, and the only rational number which is killed by the field trace is $ 0 $ (and the number in the OP is clearly nonzero).