One could do a little better
In[36]:= Clear[q, o, x]
{#, If[q = Sum[Cos[2 o x], {o, 1, \[Infinity]}, Regularization -> #];
Head[q] === Sum, "Divergent", q]} & /@ {"Abel", "Borel", "Cesaro", "Dirichlet", "Euler"}
Out[37]= {{"Abel", "Divergent"},
{"Borel", 1/2 (-Cos[2 x] - 2 Sin[x]^2)},
{"Cesaro", "Divergent"},
{"Dirichlet", -(1/2)},
{"Euler", "Divergent"}}
see the abcd[e] of divergent series. It's interesting that Cesaro's regularization is divergent in general, but
In[38]:= Sum[Cos[2 o 1/2], {o, 1, \[Infinity]}, Regularization -> "Cesaro"]
Out[38]= -(1/2)
Abel holds also a special value (the alternating sum of 1)
In[43]:= Sum[Cos[2 o \[Pi]/4], {o, 1, \[Infinity]}, Regularization -> "Abel"]
Out[43]= -(1/2)
Is it clear that there is no 0 < x < Pi
for which the conventional sum exists? This is also an input
In[47]:= Sum[Cos[2 o x], {o, 1, \[Infinity]}, VerifyConvergence -> False]
Out[47]= -(1/2)
with other words, Dirichlet.