Adding up Forever — Paradoxes at Infinity

Abstract

In Chapter 1 we mentioned sequences of rational numbers, in particular null sequences, during our discussion of the completeness property of the real number system. The main idea of this section is to say something about sequences of real numbers. We begin with a variety of examples:

• (a) 1, 2, 4, 8, 16, 32,...

• (b) 3/4, 4/5, 5/6, 6/7,...

• (c) 0, 1, −1, 0, 1, −1, 0, 1, −1,...

• (d) 1, −2, 3, −4, 5, −6,...

• (e) 1, 1, 2, 3, 5, 8, 13, 21,...

• (f) 9, −3, 1, −1/3, 1/9, −1/27, 1/81,...

• (g) 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13,...

• (h) √2 − √1, √3 − √2, √4 − √3, √5 − √4,...

• (i) (1 + 1/1)¹, (1 + 1/2)², (1 + 1/3)³,...

• (j) 1.0001, (1.0001)²/2, (1.0001)³/3,...