[racket] Puzzled with tail-recursion conversion

If you systematically apply cps and then accumulator transformations, you get this: 

(define (pi3 accuracy)
  (let helper ((k 0))
    (let ((this (formula k)))
      (if (< (abs this) accuracy)
          this
          (+ this (helper (+ k 1)))))))

(pi3 .1)

(define (pi3-cps accuracy)
  (let helper ((k 0) (c (lambda (x) x)))
    (let ((this (formula k)))
      (if (< (abs this) accuracy)
          (c this)
          (helper (+ k 1) (lambda (v) (c (+ this v))))))))

(pi3-cps .1)

(define (pi3-accu accuracy)
  (let helper ((k 0) (c 0))
    (let ((this (formula k)))
      (if (< (abs this) accuracy)
          (+ c this)
          (helper (+ k 1) (+ c this))))))

(pi3-accu .1)

Now when you compare the result of that with yours: 

(define (pi4 accuracy)
  (let helper ((k 1) (this (formula 0)))
    (if (< (abs this) accuracy)
        this
        (helper (+ k 1) (+ this (formula k))))))

you immediately see why yours fails. 

;; --- 

I would worry about accuracy because you are switching the direction of addition when you apply the accu transform, but perhaps you know why this doesn't matter for your context.