<prompt> Tafutasum_n 1infty fracn2 n - 1n 2!.  <response>: HebuS_m sum_n 1m fracn2 n - 1n 2!.Tunakokotoa hesabu chache za kwanza S_m:mpya amriarraystretch1.5anzasafuccm S_m line1 frac16 2 frac38 3 frac715 4 frac71144 5 frac419840mwishosafuTunakumbuka kuwa sehemu hizo zinaonekana kukaribia frac12, kwa hivyo pia tunakokotoa frac12 - S_m:mpya amriarraystretch1.5anzasafucccm S_m frac12 - S_m hline1 frac16 na frac13 2 frac38 frac18 3 frac715 frac130 4 frac71144 frac1144 5 frac419840 frac1840mwishosafuTunaweza kuhusisha sehemu frac12 - S_m na factorials kwa njia ifuatayo:frac13 frac23!, frac18 frac34!, frac130 frac45!, frac1144 frac56!, frac1840 frac67! .Kwa hivyo, tunakisia hiloS_m frac12 - fracm 1m 2!.Kwa hivyo, wachaT_n frac12 - fracn 1n 2!.KishaanzaalignT_n - T_n - 1 kushoto frac12 - fracn 1n 2! kulia - kushoto frac11 2 - fracnn 1! kulia fracnn 1! - fracn 1n 2! fracnn 2 - n 1n 2! fracn2 n - 1n 2!,endalignambalo ndilo hasa tunalojumlisha.Kutoka kwa utambulishofracnn 1! - fracn 1n 2! fracn2 n - 1n 2! ,tunayo hiyoanzaalignsum_n 1infty fracn2 n - 1n 2! kushoto frac12! - frac22 3! kulia kushoto frac23! - frac34! kulia kushoto frac34! - frac4 5! kulia dotsb frac12.mwishopanga 