612 Worked-Out Solutions to Exercises: Chapters 1 to 9
When we multiply these out, we get1/(−2), 1/4, 1/(−8), 1/16, 1/(−32), ...which is the same as−1/2, 1/4, −1/8, 1/16, −1/32, ...The numbers alternate between negative and positive, and their absolute values get half as
large with each repetition. This sequence is identical to the solution for Problem 2(c).
(b) If we do the same thing with a base of −1, we get(−1)−^1 , (−1)−^2 , (−1)−^3 , (−1)−^4 , (−1)−^5 , ...
This is the same as1/(−1)^1 , 1/(−1)^2 , 1/(−1)^3 , 1/(−1)^4 , 1/(−1)^5 , ...
Multiplying these out gives us1/(−1), 1/1, 1/(−1), 1/1, 1/(−1), ...which is the same as−1, 1, −1, 1, −1, ...
The numbers simply alternate between −1 and 1. This result is identical with the
solution for Prob. 2(b).
(c) Finally, let’s do the process with a base of −1/2. We get the sequence(−1/2)−^1 , (−1/2)−^2 , (−1/2)−^3 , (−1/2)−^4 , (−1/2)−^5 , ...
This is the same as1/(−1/2)^1 , 1/(−1/2)^2 , 1/(−1/2)^3 , 1/(−1/2)^4 , 1/(−1/2)^5 , ...
which is the same as1/(−1/2), 1/(1/4), 1/(−1/8), 1/(1/16), 1/(−1/32), ...
which can be simplified to−2, 4, −8, 16, −32, ...
That’s the same thing we got when we solved Problem 2(a).