TITLE: Convergent $x_n,y_n$ and $x_n^{y_n}$ diverges
QUESTION [6 upvotes]: Let $x_n, y_n > 0$. 
I need an example of convergent sequences $x_n,y_n$ such that $x_n^{y_n}$ diverges. Could you help me?

REPLY [8 votes]: Let $x_n$ be any positive sequence converging to $0$. Let $y_n$ be $1/|\ln x_n|$ if $n$ is even and let $y_n$ be $2/|\ln x_n|$ if $n$ is odd. Note that $y_n$ converges to $0$. Then $x_n^{y_n}=e^{y_n\cdot \ln x_n}$ which is $e^{-1}$ or $e^{-2}$, alternating with $n$, thus not converging.