Claim:  the sum from 0 to n of: 1/2n  is 2 - 1/2n.
Proof: (by induction)
Basis:  for n = 0 the sum is 1/20 = 1/1 = 1.
        for n = 0, 2-1/2n = 2 - 1/20 = 2-1 = 1.  
        Therefore the claim is true for n = 0.
I.H.; Assume the claim is true for some n > 0.  That is, 1 + 1/2 + 1/4 + ... + 1/2n = 2 - 1/2n.
Ind. Step:  Will show that the claim is true for n+1.
   1 + 1/2 + ... + 1/2n + 1/2n+1 = (1 + 1/2 + ... + 1/2n) + 1/2n+1
				     = (2 - 1/2n) + 1/2n+1 by I.H.
				     = 2 - 1/2n + 1/2n+1
		                     = 2 +  (-2 + 1)   = 2 - 1/2n+1 
					    ----------
					      2n+1

							
								BOX