m=1;
k=1;
γ=0.1;

DSolve[m x''[t] + γ x'[t] + k x[t] == (E^(i t)), x

(*Amplitude*)
Plot[((k-m ω^2)^2+(γ ω )^2)^(-1/2), {ω, 0, 5}, PlotRange->{0,10}]

(*Phase shift x omega (radians)*)
Plot[Pi/2 - ArcCot[γ ω / (k - m ω^2)], {ω, 0, 2}, PlotRange->{0,Pi}]

Simplify[InverseLaplaceTransform[1/((s-I ω) (m s^2 + γ s +k)), s, t]]

α = .8;
ω =  Pi/ 10;
s[x_] := Sum[(1-α)(α^t) Exp[I ω(x-t)], {t, 0, Infinity}]
Plot[Re[s[Floor[x]]], {x,0,50}]
MaxValue[Re[s[x]], x]
Abs[(1-α)/(1-α Exp[I ω])]

Plot[1/(1-Exp[-t]), {t, -10, 10}]