Disscusing convergence of a series . Let $0<a_n<1$, and $a_n\to 1,n\to \infty$.
when $\sum_{n=1}^\infty (a_n)^n$ convergent or divergent?

If $(a_n)^n=n^{-2}$, then $\sum(a_n)^n$ converges, and $a_n=n^{-2/n}\to1$. 
If $(a_n)^n=n^{-1}$, then $\sum(a_n)^n$ diverges, and $a_n=n^{-1/n}\to1$.
