using: $$\varepsilon_n(x) = \displaystyle \int_{1}^{\infty} \frac{e^{-xt}}{t^{n}}dt$$ with $n\in \mathbb{N}$. If $e_1 = \frac{1}{2};e_2=\frac{1}{8};e_3=\frac{-1}{32};e_4=\frac{-1}{128}.$
Show that: $$\varepsilon_{n}(x)\sim e^{-x}\displaystyle \sum_{s=1}^{\infty} \frac{e_s}{x^s} $$ if $x \to \infty.$ I think on doing induction over $n$, but I have trouble doing this, I would be very grateful if you can help me with this. Thanks
There are two simple methods to obtain the asymptotic series.
Option 1
The first option is to use repeated integration by parts. In the formula
$$ \int_1^\infty u\,dv = uv\Bigr|_{1}^{\infty} - \int_1^\infty v\,du, $$
take $u = t^{-n}$ and $dv = e^{-xt}{dt}$. Then repeat this on the new integral using the same pattern. Do this four times to calculate the first four coefficients $e_1,e_2,e_3,e_4$.
Option 2
Make the substitution $t = s+1$ in the integral to get
$$ \int_1^\infty e^{-xt}t^{-n} \,dt = e^{-x}\int_0^\infty e^{-xs}(s+1)^{-n} \,ds. $$
This integral is now in the standard form to which you can apply Watson's lemma.