So I was working on this problem and I got diverge, since my answer was greater than 1. The Limit was > 1, using the root test. 

$$\sum\limits\_{n=4}^\infty (1 +\frac{1}{n})^{-n^2}$$

I ended up with 

$(1+ \frac{1}{n})^{-n}$ as the limit $n$ approaches $\infty$

Using the root test. I would assume that would be diverging, but apparently it's converge. Let me know what I did wrong.