Let $\alpha \in (0,1)$ and $\delta \in (0,1/2)$ be fixed, and consider the following integrals for each integer $j \geq 0$:
$$I_j(u):= \frac{e^u}{u^{j+\alpha}} \int_{-u\delta}^0 e^t t^{j-1+\alpha}\left(1+\frac{t}{u}\right)^{-1}dt, \hspace{2mm} u>0$$
Show that for any integer $k \geq 0$ and for any integer $0 \leq j \leq k$, each of these integrals as an asymptotic development of the form
$$I_j(u) = \frac{e^u}{u^{\alpha+j}} \left( d_{0,j} + \frac{d_{1,j} }{u} + \cdots + \frac{d_{k,j}}{u^{k}} + O\left(\frac{1}{u^{k+1}}\right) \right) \text{ as }u \rightarrow \infty$$ 

**Attempts:** For $1 \leq j \leq k$, I expanded the $(1+t/u)^{-1}$ as a geometric series upto $k$ terms and got
$$I_j(u) = \frac{e^u}{u^{j+\alpha}} \left(\sum_{n=0}^k \frac{(-1)^n}{u^n} \int_{-u \delta}^0 e^t t^{j-1+\alpha+n} dt + O\left(\frac{1}{u^{k+1}}\right) \right)$$
So, I'm down to showing there exist constants $c_{0, j}, \cdots , c_{k, j}$ such that for each $0 \leq n \leq k$, 
$$\int_{-u \delta}^0 e^t t^{j-1+\alpha+n} dt = c_{n, j} + O \left( \frac{1}{u^{k+1}} \right)$$

But I'm not sure how to proceed from here (I tried integrating by parts but got too many terms...does that look promising?) or if this will work. Also will something similar work for $j=0$? Thanks.