We know $
(1+\alpha/n)^n \rightarrow e^{\alpha}
$
when $n\rightarrow +\infty$. 

Suppose we are given a modified version of the problem: 
$$
 \quad (1+\alpha\cdot a\_n)^n \tag{1}
$$
The question is to choose $a\_n$ in a way that the expression (1) has closed form when $n\rightarrow +\infty$. Can we argue that it is necessary to have $a\_n \propto \frac{1}{n}$?
Or for what families of $a\_n$ in the expression (1) is convergent?