I am working on the exercise 1.3.19 proving Stirling's formula in Stroock's "Probability Theory: an Analytic View". I need help in proving the following inequality:
\begin{equation*} [(1+\delta)e^{-\delta}]^{t-1} \int_{\delta}^{\infty}(z+1)e^{-z}dz \leqslant 2\exp{ \{1-\frac{t \delta^2}{2} + \frac{\delta^3}{3(1-\delta)} \}} \end{equation*}
Here $t \in \mathbb{R}$ is assumed to be large while $\delta \in (0,1)$. The inequality arises from estimating a part of gamma function after a change of variable.
First of all, the integral can be solved:
\begin{equation*} \int_{\delta}^{\infty}(z+1)e^{-z}dz = (\delta+2)e^{-\delta} \end{equation*}
Combine the terms, the left hand side is:
\begin{equation*} [(1+\delta)]^{t-1} (\delta+2) e^{-t\delta} \end{equation*}
It seems the right hand side might from some sort of Taylor expansion. However, the derivative seems complicated and is not giving me what I want. I am stuck here.