Shanghai, Man, a huge difference! When I tried a few polynomials, it took forever and the program crashed a few times. With your little code, it is smooth and fast! Thanks a lot. Andre

One thing can be improved for smooth animation is that to pre-compute all cases before plot in the Manipulate function:

Clear[f, fs];
f[x_,n_]:=ExpToTrig[FourierSeries[Cosh[x],x,n]]
fs=Table[f[x,n],{n,10}];
Manipulate[
Plot[
fs[[n]],{x,-3,3}],
{n,1,10,1}]

You can now drag the slider without glitch.

Generally: be careful using upper-case characters, they are very often predefined, although it works in your case. Then: Plot behaves sometimes peculiar. Try

Manipulate[Plot[Evaluate[ExpToTrig[FourierSeries[Cosh[x], x, n]]], {x, -3, 3}], {n, 1, 10, 1}]