<prompt> Pata jumla ya mfululizo sum_n0infty frac1n ! 2n cos fracpi n-12 .  <response>: Kwa x yote katika mathbbR, inashikilia beginiliyopangwa sum_n0infty fraccos leftfracpi n 2-xkulian ! xncos x sum_n0infty fraccos fracpi n2 n ! xnsin x sum_n0infty fracsin fracpi n2n ! xn cos x sum_k0infty frac-1k2 k ! x2 ksin x jumla_k0infty frac-1k2 k1 ! x2 k1cos 2 xsin 2 x1 . endaligned Hasa kwa xfrac12 tunapata jumla ya mfululizo unaohusika. Jibu: 1 