For which values of $x \in R$ does the series $\{x^\mathrm{k_n}\}^\infty_\mathrm{n=1}$ converge.
Solution 1:
Note that the series $$ \sum x^n$$ converges exactlyif $|x|<1$, and thus it also converges absolutely. This means that $\sum x^{a_n}$ as a subseries has to converge as well.
On the other hand if $|x|\geq 1$ then $|x^{a_n}|$ does not go to $0$. Thus the series cannot converge.