5 & 6

(defun abs (x)
  (if (< x 0) (* x -1) x)
    )
ABS

(defun inf_fact ()
  (let ((pprev 0) (pcur 0))
    (loop (setq pprev pcur) (setq pcur (+ 2 (/ 3.0 (+ 2.0 pcur))))
    (when (< (abs (- pcur pprev)) 0.000001) (return pcur)))
     )
  )
ABS
INF_FACT

(inf_fact)
2.645752


(defun inf_fact ()
  (let ((pprev 0) (pcur 0))
    (loop (setq pprev pcur) (setq pcur (+ 2 (/ 3.0 (+ 2.0 pcur))))
    (when (< (abs (- pcur pprev)) 0.000001) (return pcur)))
     )
  )

(defun cycle_sqrt23 (n)
  (let ((pcur 0) (i 0))
    (loop (setq pcur (/ 22.0 (+ 2.0 pcur))) (setq i (1+ i))
    (when (= i n) (return (1+ pcur))))
     )
  )
CYCLE_SQRT23

(cycle_sqrt23 40)
4.795831

(defun rec_sqrt23 (w n)
  (cond ((= n 0) (/ (+ w (/ 23.0 w)) 2))
        (T (rec_sqrt23 (/ (+ w (/ 23.0 w)) 2) (1- n)))
     )
   )
REC_SQRT23

(rec_sqrt23 23 8)
4.795832

(defun cycle_sqrt7 ()
  (let ((i 7)(pprev 1) (pcur (/ (* 2 4 6)(* 1 3 5))))
    (loop (setq pprev pcur) (setq pcur (/ (* (+ i 2.0) (+ i 4) (+ i 6))(* (+ i 1) (+ i 3) (+ i 5)))) (setq i (+ i 7))
    (when (< (abs (- pcur pprev)) 0.000001) (return pcur)))
     )
  )
CYCLE_SQRT7

(cycle_sqrt7)
1.00073