// Главный цикл
procedure TForm1.FormClick(Sender: TObject);
var x,y,s,s2:Extended; i,NN: integer ;

// Точное значение
function YY (x:Extended): Extended ;
var tmp,s: Extended ;
begin
  tmp := x/2 ;
  s := (tmp *(tmp+1) +1.0) * Exp (tmp) ;
  Result := s
end ;

//  Для заданного n
function summa (x:Extended;n:integer): Extended ;
var i,nf: integer; tmp,tmp1,s: Extended ;
begin
  s := 1.0 ;
  tmp := x/2 ;
  tmp1 := 1.0 ;
  nf := 1 ;
  for i := 1 to n do
    begin
      nf := nf * i ;
      tmp1 := tmp1*tmp ;
      s := s + (i*i+1) * tmp1 / nf ;
    end ;
  summa := s;
end ;

//  Для заданного epsylon
function summa2 (x,eps:Extended;var nitt:integer): Extended ;
var i,nf: integer; eps1,tmp,tmp1,s,slast: Extended ;
begin
  nitt := 0 ;
  eps1 := 1 ;
  slast := 1 ;
  s := 1.0 ;
  tmp := x/2 ;
  tmp1 := 1.0 ;
  nf := 1 ;
  i := 1 ;
  while (Abs(eps1)>eps)do
    begin
      Inc (nitt) ;
      nf := nf * i ;
      tmp1 := tmp1*tmp ;
      s := s + (i*i+1) * tmp1 / nf ;
      eps1 := s - slast ;
      slast := s ;
      Inc (i) ;
    end ;
  summa2 := s;
end ;

begin
   for i := 1 to 10 do
   begin
     x := i/10.0 ;
     y := YY (x) ;
     s := summa (x,10) ;
     s2 := summa2 (x,0.00001,NN) ;
     Memo1.Lines.Add(Format('%3.1f  %9.6f %9.6f %9.6f %6d',[x,y,s,s2,NN]))
   end ;
end;