Clear[PerEx,ff,aa,bb,x];PerEx[x_,ff_,aa_,bb_]:=ff[x-(Ceiling[((x-aa)/(bb-aa))]-1)(bb-aa)]

#&[4]

Plot[PerEx[x,((#)^2)&,-2,1],{x,-4,4},PlotRange->{-1,5}]

aa[0]=Sinh[1]

aa[n_]=(((((-1))^n)2 Sinh[1])/(1+(((n Pi))^2)))

bb[n_]=(((((-1))^(n+1))n Pi 2 Sinh[1])/(1+(((n Pi))^2)))

Plot[PerEx[x,Exp[#]&,-1,1],{x,-4,4},PlotRange->{-1,5}]

Plot[{PerEx[x,Exp[#]&,-1,1],aa[0]+Sum[aa[k] Cos[k Pi x]+bb[k] Sin[k Pi x],{k,1,10}]},{x,-4,4},PlotRange->{-1,5}]

Plot[{PerEx[x,Exp[#]&,-1,1],aa[0]+Sum[aa[k] Cos[k Pi x]+bb[k] Sin[k Pi x],{k,1,130}]},{x,-1,1},PlotRange->{0,3}]

{Exp[x],(aa[0]+Sum[aa[k] Cos[k Pi x]+bb[k] Sin[k Pi x],{k,1,140}])}/.{x->0.9999}

{Exp[x],(aa[0]+Sum[aa[k] Cos[k Pi x]+bb[k] Sin[k Pi x],{k,1,40}])}/.{x->0.01}

Plot[{PerEx[x,Cosh[#]&,-1,1],aa[0]+Sum[aa[k] Cos[k Pi x],{k,1,10}]},{x,-4,4},PlotRange->{0,2}]

Plot[{PerEx[x,Sinh[#]&,-1,1],Sum[bb[k] Sin[k Pi x],{k,1,10}]},{x,-4,4},PlotRange->{-2,2}]