# Exercise 1.38 of SICP

Exercise 1.38: In 1737, the Swiss mathematician Leonhard Euler published a memoir De Fractionibus Continuis, which included a continued fraction expansion for e - 2, where e is the base of the natural logarithms. In this fraction, the Ni are all 1, and the Di are successively 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, …. Write a program that uses your cont-frac procedure from exercise 1.37 to approximate e, based on Euler’s expansion.

Since all the continued fraction works is done from the previous exercise the trickiest part is defining a function for the Di’s. Procedure d should always return 1 except for the special indices. For me it was key to notice that every 3i-1 index, where is is i=1…n, was the even number I care about. And that even number was of the form 2i.

(define n (lambda (i) 1.0))
(define (d i)
(if (= 0 (remainder (+ i 1) 3))
(* 2 (/ (+ i 1) 3))
1))

(define (cont-frac n d k)
(define (cf i)
(if (= i (+ k 1))
0
(/ (n i)
(+ (d i) (cf (+ i 1))))))
(if (not (> k 0))
0
(cf 1)))

(define (cont-frac-iter n d k)
(define (iter k result)
(if (= k 0)
result
(iter (- k 1) (/ (n k) (+ (d k) result)))))
(iter k 0))

> (cont-frac n d 11)
.7182818352059925

> (cont-frac-iter n d 11)
.7182818352059925


$$e-2 = 0.7182818284590452353$$

$$\epsilon = |e-2-0.7182818352059925| = 0.0000000067469472647$$