T = 10
ω = 2*Pi/T

f1 = Exp[-t]

Plot[Evaluate[f1], {t, 0, T}]

a0 = (2/T)*(∫_0^T) f1*
            Cos[0*ω*t] \[DifferentialD]t

a1 = (2/T)*(∫_0^T) f1*
            Cos[1*ω*t] \[DifferentialD]t

a2 = (2/T)*(∫_0^T) f1*
            Cos[2*ω*t] \[DifferentialD]t

b1 = (2/T)*(∫_0^T) f1*
            Sin[1*ω*t] \[DifferentialD]t

a3 = (2/T)*(∫_0^T) f1*
            Cos[3*ω*t] \[DifferentialD]t

b2 = (2/T)*(∫_0^T) f1*
            Sin[2*ω*t] \[DifferentialD]t

b3 = (2/T)*(∫_0^T) f1*
            Sin[3*ω*t] \[DifferentialD]t

f2 = 
      a0/2  +  a1*Cos[ω*t] + b1*Sin[ω*t] + 
        a2*Cos[2*ω*t] + b2*Sin[2*ω*t] + a3*Cos[3*ω*t] + 
        b3*Sin[3*ω*t]

Plot[Evaluate[f2], {t, 0, T}]