(* Parameters chosen by class: Period is 7. Function f=t^4, but odd so that from -3.5 to 0 f=-t^4. *)

w0=2 Pi/7;
b[n_]= 2/7 ( Integrate[-(t^4) Sin[n w0 t],{t,-3.5,0}]   +    Integrate[t^4 Sin[n w0 t],{t,0,3.5}] ) (*Need to integrate piecewise because f(t) changes from -t^4 to t^4 *)
Table[b[n],{n,1,20}];  (*looking at the first 20 coefficients *)
%//TableForm

(*Let's see how well the series does at reproducing the specified function!*)
f[t_]=Sum[b[n] Sin[n w0 t], {n,1,100}];
Plot[f[t],{t,-14,14}, PlotRange->All]
Plot[f[t],{t,-1,1}, PlotRange->All] (*zooming in to see wiggles *)

f[t_]=Sum[b[n] Sin[n w0 t], {n,1,10000}];  (*more terms to reduce the wiggles *)
Plot[f[t],{t,-1,1}, PlotRange->All]

Plot[f[t],{t,3.4,3.6}, PlotRange->All]  (*Gibbs phenomena when sharp transitions *)