A0 = 1 - 1/E
A2 = 25/2 - 35/E

f[x_] = A0 LegendreP[0,x] + A2 LegendreP[2,x]

Plot[f[x],{x,-1,1}]

g[x_] = Exp[-Abs[x]];
A[l_] :=(2l +1)/2 Integrate[g[x] LegendreP[l,x],{x,-1,1}]

h[x_] = Sum[A[l]LegendreP[l,x],{l,0,5}]

Plot[h[x],{x,-1,1}]