Answers for Chapter 3 Sample Problems


1. Ian's problem

2. Ian's proof that Infinity = -1

First Ian found the sum of the infinite series

1 +a + a2 + a3 + a4 + a5 + ... = C (which he called C).

 He factored an 'a' out of each term
 
	1 + a(1 + a + a2 + a3 + a4 + a5...) = C
			    but what's in parentheses is C
		
         So    1 + a(C) = C     now he solved for C
				he subtracted C and 1 from 
				both sides		

		aC - C = -1  	he then factored out C


		C(a-1) = -1  	he then divided by a-1


		C = -1/(a-1) 	he then mult. top and bottom by -1

So
       1 + a + a2 + a3 + a4 + a5 + ... = 1/(1-a)

Now Ian put 2 in for a and got
        	1 + 2 + 4 + 8 + 16 + 32 ... = 1/(1-2) = -1
Since the left side goes to infinity, he concluded that

		Infinity = -1

What's wrong with Ian's argument? Newton and Euler and those other mathematicians had made similar mistakes with infinite series! The problem is that the infinite series converges only when the absolute value of a is less than 1 . So it's OK to make mistakes ..if you start early!
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