   Chapter 11.2, Problem 81E

Chapter
Section
Textbook Problem

What is wrong with the following calculation? 0 = 0 + 0 + 0 + ... = ( 1 − 1 ) + ( 1 − 1 ) + ( 1 − 1 ) + ... = 1 − 1 + 1 − 1 + 1 − 1 + ... = 1 + ( − 1 + 1 ) + ( − 1 + 1 ) + ( − 1 + 1 ) + ... = 1 + 0 + 0 + 0 + ... = 1 (Guido Ubaldus thought that this proved the existence of God because “something has been created out of nothing.”)

To determine

To check:

What’s wrong with the given calculation?

Explanation

1) Concept:

The geometric series is

n=1arn-1=a+ar+ar2+,

If r<1 then the geometric series is convergent

If r1 then the geometric series is divergent

2) Given:

0=0+0+0+

=1-1+1-1+1-1+

=1-1+1-1+1-1+<

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