f=1/((1+x^2)(4-x^2))

f//Apart

alpha = 0.2

F[x_] := (Sin[alpha*Pi]/Pi)(1/alpha)+Sum[(((-1)^n)*2*alpha*Sin[alpha*Pi]Cos[n*x])/(alpha^2-n^2),{n,1,5}]

Plot[F[x],{x,-2Pi,2Pi},PlotPoints->200]

s=((-1^n)*2*alpha*Sin[alpha*Pi]Cos[n*x])