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.
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