Pensieve header: Games with asymptotic series.

Assuming[ϵ>0,Integrate[((E^(-x))/(1+ϵ x)),{x,0,∞}]]

Series[Assuming[ϵ>0,Integrate[((E^(-x))/(1+ϵ x)),{x,0,∞}]], {ϵ,0,3}]

Assuming[ϵ>0,Series[Integrate[((E^(-x))/(1+ϵ x)),{x,0,∞}],{ϵ,0,7}]]

Plot[Integrate[((E^(-x))/(1+ϵ x)),{x,0,∞}], {ϵ,0,2}]

Plot[-(((E^(1/ϵ)) ExpIntegralEi[-(1/ϵ)])/ϵ),{ϵ,0,2}]

NIntegrate[((E^(-x))/(1+ϵ x))/.ϵ->0.1,{x,0,∞}]

Normal[Assuming[ϵ>0,Series[Integrate[((E^(-x))/(1+ϵ x)),{x,0,∞}],{ϵ,0,7}]]]/. ϵ->0.1

Table[Sum[(((-ϵ))^k)k!, {k,0,n}],{n,0,30}]/.ϵ->0.1

N[Integrate[((E^(-x))/(1+ϵ x))/.ϵ->1/157,{x,0,∞}],100]

Table[Sum[N[(((-ϵ))^k)k!,157], {k,0,n}],{n,0,3*157,6}]/.ϵ->1/157

fs={-(((E^(1/ϵ)) ExpIntegralEi[-(1/ϵ)])/ϵ)}~Join~Table[Normal@Assuming[ϵ>0,Series[Integrate[((E^(-x))/(1+ϵ x)),{x,0,∞}],{ϵ,0,k}]],{k,{1,5,25,125}}];
Plot[fs,{ϵ,0,1},PlotPoints->1000,PlotRange->{0,1},PlotLegends->{f}~Join~Table[k,{k,{1,5,25,125}}]]