Example on Instability

e:= E//N

G[0]=e -1; G[n_]:= e - n * G[n-1]

F[n_]:= Integrate[x^n Exp[x], {x, 0,1}]

Do[Print[StringForm["n=``,         F =``,              G=``", 
n , F[n]//N , G[n] ]], {n, 25}]

Integrate[x^(20) Exp[x], {x, 0,1}]

H[n_]:=NIntegrate[x^(n) Exp[x], {x, 0,1}] ;
Do[Print[StringForm["n=``,      H=``", n, H[n] ] ],   
{n, 16,25}]