b[n_] := (4/((2*k + 1)*Pi));
f[x_, 
          N_] := (∑_(k = 0)^N) b[n]*
            Sin[(2*k + 1)*x];
Plot[f[x, 20], {x, -3*Pi, 3*Pi}]

b[k_] := (4/((2*k + 1)*Pi));
f[
          x_] := (∑_(k = 0)^∞) b[k]*
            Sin[(2*k + 1)*x];
(∫_0^(2*Pi))f[x]^2 \[DifferentialD]x

2 π
b[k_] := (4/((2*k + 1)*Pi));

Pi*(∑_(k = 0)^∞) b[k]^2

2 π
c[_k] := (I/((2*k + 1)*Pi))*(Cos[2*k*Pi + Pi] - 
              1);
c[_ - 
            k] := ((-I)/((2*k + 1)*Pi))*(Cos[
                2*k*Pi + Pi] - 1);
2*Pi*(∑_(k = -∞)^∞) c[k]*
          c[-k]

2 π (∑_(k = -∞)^∞) 
            c[-k] c[k]

f[t_] := 
        Which[t > 0, (E^(-t)), 
          t < 0, (E^t)];
F[w_] := (∫_(-∞)^∞) f[
            t]*(E^(-I*w*
                t)) \[DifferentialD]t
\(∫\_\(-∞\)\%∞\( f[
              t]\^2\) \(\(\[DifferentialD]\)\(t\)\(	\)\)\)

n = Abs[F[w]]

(1/(2*Pi))*(∫_(-∞)^∞) 
              (n^2) \[DifferentialD]w