Series when differential operator involved – mathematica.stackexchange.com

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 ...