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/* 
 
It is approximated by finite continued  fractions :

[0;1,1,1,....]

https://commons.wikimedia.org/wiki/File:Finite_continued_fractions_0;1,1,1,.....png

*/

kill(all);


a:[0,3,2,1000,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1];/* continuead fraction - golden mean */



/* floating point value of n-th convergent */
f(i):=block
(
	b : firstn(a,i), /* first n terms of a */
	c : cfdisrep(b),
	c: ratsimp(c),
	print(c), 
	float(c)
)$




iMax : length(a);


/* save the values to 2 lists */
xx:makelist (i, i, 1, iMax); /* list of positive integers  */
yy:makelist (f(i), i, 1, iMax); /* list of cf  */

/*
for i:1 thru iMax step 1 do 
(  
   xx:cons(i,xx),  
   y:float(f(i)),
   yy:cons(y,yy)

);
*/
load(draw);
draw2d(
   file_name = "g700",
   terminal  = 'png,
  
  
   dimensions  = [700,700],
   yrange = [0.0,0.35],
   xrange = [0, iMax+1],
   title= "Finite continued fraction aproximation for [0,3,2,1000,1,...] = .2857346851349422",
   key = "nth-covergent",
   xlabel     = "n",
   ylabel     = "n-continued fractions",
   point_type    = filled_circle,
   point_size    = 1.0,
   points_joined = true,

   color         = red,
   points(xx,yy),
   color = blue,
   key = "[0,3,2,1000,1,...]",
   explicit(0.2857346725405882,x,1,iMax)

  );

Text output:

 batch("c.mac");

read and interpret file: #p/home/a/maxima/batch/cf/limit/c.mac
(%i9) kill(all)
(%o0)                                done
(%i1) a:[0,3,2,1000,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
(%o1) [0, 3, 2, 1000, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
(%i2) f(i):=block(b:firstn(a,i),c:cfdisrep(b),c:ratsimp(c),print(c),float(c))
(%i3) iMax:length(a)
(%o3)                                 21
(%i4) xx:makelist(i,i,1,iMax)
(%o4) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 
                                                                            21]
(%i5) yy:makelist(f(i),i,1,iMax)
0 
1
- 
3
2
- 
7
2001
---- 
7003
2003
---- 
7010
4004
----- 
14013
6007
----- 
21023
10011
----- 
35036
16018
----- 
56059
26029
----- 
91095
42047
------ 
147154
68076
------ 
238249
110123
------ 
385403
178199
------ 
623652
288322
------- 
1009055
466521
------- 
1632707
754843
------- 
2641762
1221364
------- 
4274469
1976207
------- 
6916231
3197571
-------- 
11190700
5173778
-------- 
18106931
(%o5) [0.0, 0.3333333333333333, 0.2857142857142857, 0.2857346851349421, 
0.2857346647646219, 0.2857346749446942, 0.2857346715502069, 
0.2857346729078662, 0.2857346723987228, 0.2857346725945441, 
0.2857346725199451, 0.2857346725484682, 0.2857346725375775, 0.285734672541738, 
0.2857346725401489, 0.2857346725407559, 0.2857346725405241, 
0.2857346725406126, 0.2857346725405788, 0.2857346725405917, 0.2857346725405868]
(%i6) load(draw)
(%o6)            /usr/share/maxima/5.41.0/share/draw/draw.lisp
(%i7) draw2d(file_name = "g700",terminal = 'png,dimensions = [700,700],
             yrange = [0.0,0.35],xrange = [0,iMax+1],
             title = "Finite continued fraction aproximation for [0,3,2,1000,1,...] = .2857346851349422",
             key = "nth-covergent",xlabel = "n",
             ylabel = "n-continued fractions",point_type = filled_circle,
             point_size = 1.0,points_joined = true,color = red,points(xx,yy),
             color = blue,key = "[0,3,2,1000,1,...]",
             explicit(0.2857346725405882,x,1,iMax))
(%o7)                      [gr2d(points, explicit)]
(%o7)                                c.mac
(%i8)