Rational Series That Sum to an Irrational Number

Date: 07/06/98 at 21:11:30
From: Mario Carvajal
Subject: Pi

How can the sum of an infinite series of rational numbers result in an
irrational number? For example, the sums of certain series contain pi.

Thank you for your answer.

Date: 07/07/98 at 11:23:59
From: Doctor Rob
Subject: Re: Pi

The technical reason is that the rational numbers are not "complete"; 
that is, they do not contain all the limits of all sequences of 
rational numbers. Here is another example:

   1, 3/2, 7/5, 17/12, 41/29, 99/70, ...

Each number is derived from the previous number by adding one, taking 
the reciprocal, and then adding one again. (There are also other easy 
rules for how to get a term from the preceding one.) The limit of this 
sequence is sqrt(2), which is irrational. You can prove that the 
first, third, fifth, and every odd-numbered term in this sequence is 
less than sqrt(2), and every even-numbered one is greater than 
sqrt(2), yet the distance between successive terms is decreasing to 0. 
Thus this sequence has no rational number as its limit.

It is because of this property that the real numbers were invented.  
They are defined as the set of limits of all sequences of rational 
numbers (more or less).

- Doctor Rob, The Math Forum
Check out our web site! http://mathforum.org/dr.math/