Rational Series That Sum to an Irrational NumberDate: 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/ |
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