Summe = 1
K = 0
\(While[
      1/\(\((K + 1)\)!\)\ \((K + 2)\)/\((K + 1)\) > \ 
        10^\((\(-150\))\), 
\(K++\); 
Summe\  += 1/\((\(K!\))\); 
 (*\ 
        Summe\  == \ 
          Sum[1/\(k!\), {k, 0, K}]\ *) 
Print["K=", 
        K, ", Fehlerabschätzung = ", \ 
        N[1/\(\((K + 1)\)!\)\ \((K + 2)\)/\((K + 
                  1)\)]]
]\)