$$\sum_{n=1}^\infty \left(n^{n^a}-1\right)$$
Plugging in numbers on wolfram alpha I know that this sum converges when a is strictly less than -1. It is kinda very similar to the p-series test because we literally have a p-series 1/n^p in the power of n when p=-a and a<0. But I need to know WHY the sum converges WHEN a<-1. I can prove all the situations when the sum Diverges when a is greater or equal to -1 except this one. Picture of the sum
