Series when differential operator involved – mathematica.stackexchange.com
metroidman
f[n_Integer] := Nest[-D[#, x]/2 &, Cos[x], n]
Sum[f[n], {n, 0, Infinity}](*does not work*)
The sum of that series can be manually found as
$$s(x) = \frac{4}{5} \cos x + \frac{2}{5} \sin x$$
Is ...