Chapter 8
- When a negative number is raised to an even positive integer power, the result is always
a positive number. When a negative number is raised to an odd positive integer power,
the result is always a negative number. - The answers, along with explanations, are as follows.
(a) If we raise a base of −2 to increasing integer powers starting with 1, we get this sequence:
(−2)^1 , (−2)^2 , (−2)^3 , (−2)^4 , (−2)^5 , ...When we multiply these out, we get−2, 4, −8, 16, −32, ...The numbers alternate between negative and positive, and their absolute values double
with each repetition. This sequence “runs away” toward both “positive infinity” and
“negative infinity”!
(b) If we do the same thing with a base of −1, we get(−1)^1 , (−1)^2 , (−1)^3 , (−1)^4 , (−1)^5 , ...Multiplying these out gives us−1, 1, −1, 1, −1, ...The numbers simply alternate between −1 and 1.
(c) If we carry out the same process with a base of −1/2, we get(−1/2)^1 , (−1/2)^2 , (−1/2)^3 , (−1/2)^4 , (−1/2)^5 , ...Multiplying these out produces−1/2, 1/4, −1/8, 1/16, −1/32, ...The numbers again alternate between negative and positive, and their absolute values get
half as large with each repetition. This sequence converges toward 0 “from both sides.”- Here are the answers. Note how they “mirror” the results of Prob. 2.
(a) If we raise a base of −2 to smaller and smaller negative integer powers starting with
−1, we get this sequence:
(−2)−^1 , (−2)−^2 , (−2)−^3 , (−2)−^4 , (−2)−^5 , ...
This is the same as1/(−2)^1 , 1/(−2)^2 , 1/(−2)^3 , 1/(−2)^4 , 1/(−2)^5 , ...Chapter 8 611