In[]:=
f[t_]=Sum[Sin[(2n+1)t]/(2n+1),{n,0,∞}]
Out[]=
1
2
(ArcTanh[
-t

]-ArcTanh[
t

])
In[]:=
Plot[f[t],{t,-5,5}]
Out[]=
-4
-2
2
4
-0.5
0.5
In[]:=
Plot[Sum[Sin[(2n+1)t]/(2n+1),{n,0,15}],{t,-5,5}]
Out[]=
-4
-2
2
4
-1.0
-0.5
0.5
1.0
In[]:=
f[a_,t_]=​​Sum[Sin[(2n+1)t]/(2n+1)Exp[-an],{n,0,∞},Assumptions{a>0}]
Out[]=
1
2

a/2

ArcTanh
-
a
2
-t

-ArcTanh
-
a
2
+t


In[]:=
Plot[f[1,t]//Re,{t,-5,5}]
Out[]=
-4
-2
2
4
-0.5
0.5
In[]:=
Plot[f[0.1,t]//Re,{t,-5,5}]
Out[]=
-4
-2
2
4
-0.5
0.5