Notations :
Let $\alpha >0$ be a positive real number and let's consider $\phi : \mathbb R^+ \to \mathbb N^*$ defined as follow: $$ \phi(x) = \lfloor e^{\alpha} x^\alpha\rfloor $$
We note $f : x \mapsto \sum_{n=1}^{\phi(x)}n^{-n/\alpha}x^n$ for $x\geq 0$.
The problem:
Find an equivalent of $f(x) = \sum_{n=1}^{\phi(x)}n^{-n/\alpha}x^n$ as $x\longrightarrow +\infty$.
What I tried:
I tried to use the summation by parts formula but that but cannot evaluate the new expressions properly. I also tried to compare with an integral but the expression is still hard to evaluate.