456 Chapter 8 • Infinite Series of Real Numbers
(^00 1 1 1 1)
Example 8.1.6 The harmonic series '""' - = 1 + -+ -+ · · · + - + · · ·.
n=l Ln^2 3 n
00 1
In Example 2.5.16 we showed that this series diverges; in fact, L - = +oo.
n=l n
0
00
1 1 1 1
Example 8.1.7 The p-series '""' Ln-P = 1 + - + - + 2P 3P · · · + - + nP · · ·.
n=l
Convergence or divergence of this series depends on the value of p. We shall
have more to say about this series in the next section. 0
00
Example 8.1.8 Given any sequence {bn}, the series L (bn - bn+i) is called
n=l
a telescoping series because of the telescoping (or collapsing) nature of its
partial sums:
n
Sn = L (bk - bk+l) (4)
k=l
= b1 - bn+l·
Thus, a telescoping series (4) converges if and only if the sequence {bn}
co nverges. (See Exercises 7- 14.) 0
00
Definition 8.1.9 (Grouping by Inserting Parentheses) Let L an be an
n=l
infinite series, and let { nk} be a strictly increasing sequence of positive integers.
Define the sequence {bk} by
b1 = a1 + a2 + a3 + · · · + an 1
b2 = an, +l + an 2 +2 + an 3 +3 + · · · + an2
b3 = an 2 +1 + an 2 +2 + an 2 +3 + · · · + an 3
00 00
Then L bn is said to be a grouping of the series L an by inserting
n=l · n=l
parentheses.