Tuesday, October 4, 2011

Implementing cauchy sequences

I have decided to work on constructing some of the computable numbers using cauchy sequences. Generally, this is as simple as mapping over the counting numbers. Here is a construction of e:

(def e
  (fn [i]
    (apply + (map (comp / factorial) (range i))))

You can evaluate this sequence into:

(0 1 2 5/2 8/3 65/24 163/60 1957/720 ...)

In this way we will provide us with an approximation for e. Here are some additional sequences:

(def pi
  (fn [n]
    (/ (* 4 (Math/pow -1 n))
       (+ (* 2 n) 1)))))

(def golden-ratio
  (fn [i]
    (/ (nth fib (inc i))
       (nth fib i)))
  (rest (range))))

These should make for relatively good approximations of these irrational numbers.

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