(*  ML ASSESSED EXERCISES. TICK 2* SUBMISSION *)

(* PROBLEM 1.1 - A recursive function to sum 1+1/2+1/8... *)

fun sumtr(n)=
  let
    (* this is the recursive function *)
    fun f(term,m)=
      if m=0 then
        0.0
      else
        term+f(term/2.0,m-1)
  in
    f(1.0,n)
  end;

(* PROBLEM 1.2 - An iterative function to sum 1+1/2+1/4... *)

fun sumti(n)=
  let
    fun f(total:real,term:real,m)=
      if m=0 then
        total
      else
        f(total+term,term/2.0,m-1)
  in
    f(0.0,1.0,n)
  end;

(* PROBLEM 2 - An iterative function to approximate e *)

fun eapproxi(n)=
  let
    fun f(total:real,term:real,m,mup:real)=
      if m=0 then
        total
      else
        f(total+term,term/mup,m-1,mup+1.0)
  in
    f(0.0,1.0,n,1.0)
  end;

(* PROBLEM 3 - An iterative function to compute exponentials *)

fun expi(z:real,n)=
  let
    fun f(total:real,term:real,m,mup:real,y:real)=
      if m=0 then
        total
      else
        f(total+term,term*y/mup,m-1,mup+1.0,y)
  in
    f(0.0,1.0,n,1.0,z)
  end;