Why Read This Book? - Index of

88 Chapter 3 Sets and Their Properties
If x �= 1, we may divide both sides through by 1 − x to have
1 + x + x 2 + x 3 + x 4 =
1 − x5
1 − x
This algebraic observation suggests a general formula for the sum of powers of a
real number x �= 1. If n ≥ 0, then
1 + x + x 2 + x 3 +···+x n =
n�
x k =
k=0
Prove Eq. (3.51) with an induction argument on n ≥ 0.
1 − xn+1
1 − x
(3.51)
EXERCISE 3.5.11 Prove the following factorization formula for nonzero real
numbers a and b and nonnegative integer n. 18
a n+1 − b n+1 = (a − b)(a n + a n−1 b + a n−2 b 2 +···+a 2 b n−2 + ab n−1 + b n )
(3.52)
Another reason Theorem 3.5.1 is useful is that a result might not be true for all
positive integers, but only eventually true, that is, true for n ≥ j for some positive
integer j. Just like we use � n k=1 ak to denote the sum of the ak, we can use the
notation � n k=1 ak to denote the product of the terms. In other words,
� n k=1 ak = a1 · a2 · a3 ···an
EXERCISE 3.5.12 For all n ≥ 2, �n � � 1 1
k=2 1 − k = n .
EXERCISE 3.5.13 If n ≥ 4 is an integer, then 2 n 2n + 1.
Proof. Let n ≥ 3 be an integer. Then
n 2 = n · n ≥ 3n = 2n + n>2n + 1
EXERCISE 3.5.16 If n ≥ 5 is an integer, then n 2 < 2 n .
18 Don’t make another induction argument. Letting x = b/a in Eq. (3.51) provides a good start.