The series is
$\sum_{n=0}^{\infty}\left(1+\frac{n}{a}\right)^bx^n,$
where $a$ and $b$ are positive real numbers, $x\in[0,1]$.
The sum of the series diverges when $x\to1$. I want to get an upper bound of this sum, and I found an exisiting result (given by @Sangchul Lee) which seems very helpful
$\sum_{n=0}^{\infty}n^kx^n\leq \frac{k!}{(1-x)^{k+1}}.$
Unfortunately, the author did not give the proof of the result so I have no idea how to apply this result to my problem.