# Function as a sum of functions

I just get started with Mathematica and I have a basic question about defining a function (I am sorry if this was asked before). I have 2 functions defined as:

lorentz[x_, x0_, linewidth_] := 1/Pi (1/2 linewidth)/((x - x0)^2 + (1/2 linewidth)^2) ;
mol[J_, dB_] := dB * J (J + 1);


I want to create a new function which combines the 2 functions above, as follows. For example in Python (that's what I used so far) I would do this:

linewidth = 0.1
dB = 0.1
def f_combined(x):
f = 0
for J in range(1,10):
f = f + lorentz(x,mol(J,dB),linewidth)
return f


I am not sure how to do it in Mathematica. This is what I tried:

linewidth = 0.1;
dB = 0.1;
f_combined[x_] = Sum[lorentz[x, mol[J, dB], linewidth] , {J, 1, 10}];


But it doesn't seem to work. Can someone help me with this? Thank you!

Blank ( _ ) is only used as a pattern object; it cannot be used in a variable name

Clear["Global*"]

lorentz[x_, x0_, linewidth_] :=
1/Pi (1/2 linewidth)/((x - x0)^2 + (1/2 linewidth)^2);
mol[J_, dB_] := dB*J (J + 1);

linewidth = 1/10;
dB = 1/10;
fCombined[x_] =
Sum[lorentz[x, mol[J, dB], linewidth], {J, 1, 10}] // Together //
FullSimplify;

Plot[fCombined[x], {x, -1, 15}]


A slight variation using the Fold function (benefitting from Bob Hanlon's answer):

lorentz[x_, x0_, linewidth_] := 1/Pi (1/2 linewidth)/((x - x0)^2 + (1/2 linewidth)^2) ;
mol[J_, dB_] := dB * J (J + 1);
linewidth = 0.1;
dB = 0.1;

fCombined[x_] =
Fold[#1 + lorentz[x, mol[#2, dB], linewidth] &, 0, Range[10]] //
Together // FullSimplify


The Fold function starts at 0, folds one argument (1-10) representing an increasing J every time it executes the expression defined as the first argument, thus accumulating a sum in terms of x.

Plot[fCombined[x], {x, -1, 15},
PlotStyle -> {Thin, Blue},
GridLines -> Automatic,
GridLinesStyle -> {{Gray, Dotted}, {Gray, Dotted}},
PlotRange -> {{-1, 13}, {0, 7.0}},
AspectRatio -> 0.25,
AxesLabel -> {x, Defer@fCombined[x]},
ImageSize -> 800]
`