f[x_]:=E^(-Pi x^2); (* Not E^(-2 Pi x^2) !... *)

Plot[f[x],{x,-2,2}]

partialSum[n_,t_]:=Sum[f[m/t],{m,-n,n}];

partialSum[200,100]//N

partialSum2[n_,t_]:=t Sum[f[m t],{m,-n,n}];

partialSum2[1,100]//N

ListPlot[{Table[partialSum[n,100],{n,1,200}],Table[partialSum2[n,100],{n,1,200}]}]