title: Converting a repeating binary number to decimal (express as a series?)
body: <p>Given a binary number that repeats, for example 0.(0011) or 0.0(101), how would one go about converting it to decimal?</p>
  
  <p>What I've been able to dig up so far is the simple method for converting a terminating binary number to decimal, as below:</p>
  
  <pre><code>res(N+2) = res(N+1) / 2 + res(N)
  </code></pre>
  
  <p>where res is the result after step N, and N is the current iteration (N=0; n->(num binary digits)). Applying that repeatedly to a nonterminating binary number gives a good approximation, for example</p>
  
  <pre><code>dec:0.4 || bin: 0.(0110):
  
  0     / 2 + 0 = 0
  0     / 2 + 0 = 0
  0     / 2 + 1 = 1
  1/2   / 2 + 1 = 3/2
  3/2   / 2 + 0 = 3/4
  3/4   / 2 + 0 = 3/8
  3/8   / 2 + 1 = 19/16
  19/16 / 2 + 1 = 51/32
  51/32 / 2 + 0 = 51/64
  51/64 / 2 + 0 = 51/128 = 0.3984
  </code></pre>
  
  <p>which is approximately 0.4.</p>
  
  <p>So, I've got a means of calculating approximates, but I'm struggling with finding a way to express this. I've started trying to write it as a series I can compute at the limit as n->inf without too much success so far.</p>