Discrete Mathematics Demystified

270 Discrete Mathematics Demystified
Solution: The terms c j = 17− j certainly tend to zero. So the series might converge,
but the zero test will not tell us. In Example 13.9, we in fact learned that the
series does converge.
13.8 Basic Properties of Series
We now present some elementary properties of series that are similar to the properties
of sequences given in Theorem 12.1.
Theorem 13.2 Suppose that ∞ D. Then
j=1 c j converges to C and ∞
1. ∞
j=1 (c j ± d j) converges to C ± D.
2. ∞
j=1 αc j converges to α · C, where α is any real constant.
j=1 d j converges to
Corollary 13.1 If ∞ j=1 b j diverges and α is any nonzero constant then ∞ j=1 α · b j
diverges.
Proof of Corollary 13.1: Seeking a contradiction, we suppose that ∞
j=1 α · b j
converges. By Part (2) of the theorem,
∞
j=1
also converges. That is a contradiction.
1
α · α · b j =
∞
b j
j=1
Here are some examples which indicate how Theorem 13.2 and its corollary can
be applied.
EXAMPLE 13.16
Discuss convergence for the series
∞
j=1
Solution: We know by Example 13.3 that
∞
j=1
4
2 j
1
2 j
(13.2)