Chapter 7
Infinite Series
7.1
Sequence
Mathematicslly, a sequence is defined as a function whose domain is the set of positive integrals. Although a sequence is a function, it is common to represent sequence subscript notation rather than by the standard function notation. For instance, in the sequence
1 2 3 4 ↓ ↓ ↓ ↓ ↓ 1 2 3 4
Example 170 (a) The term of the sequence {} = {3 + (−1)} are
3 + (−1) 3 + (−1)2 3 + (−1)3 3 + (−1)4 2 4 2 4
(b) The term of the sequence {} =
© 2 1+ ª are 2·1 1+1 2·2 1+2 2·3 1+3 2·4 1+4 2 2 4 3 6 4 8 5
(c) The term of the sequence {} =
n 2 2−1 o are 12 21−1 22 22−1 23 23−1 24 24−1 1 1 4 3 9 7 16 15 94
Limit of a Sequence:
Definition 37 Let be a real number. The limit of sequence {} is
written as
lim
→∞ =
if for each 0 there exists 0 such that |− | whenever
Sequence that have limits converge, whereas sequence sequences that do not have limits diverge.
Theorem 76 Let be a function of a real variable such that lim
→∞ () =
If {} is a sequence such that () = for every positive integer then
lim
→∞=
Example 171 Find the limit of the sequence whose th term is =
¡
1 +1¢ Theorem 77 (Properties) Let
lim
→∞= and →∞lim =
() lim
→∞(± ) = ± () lim→∞ = is any real number
() lim →∞() = () lim→∞ = 6= 0 and 6= 0
Example 172 () Because the sequence {} = {3 + (−1)} has terms
2 4 2 4 that alternate between 2 and 4 the limit
lim
→∞
does not exists. Thus, the sequence diverges. () For {} =
©
1−2
ª
you can divide the numerator and denominator by to obtain lim →∞ 1− 2 = lim→∞ 1 (1)− 2 =− 1 2 which implies that the sequence converges to −12
Example 173 Show that the sequence whose th term is =
2
2−1
con-verges.
Proof. Consider the function of real variable () =
2
2− 1
Applying L’Hopital Rule.
Theorem 78 (Squeeze Theorem) If lim
→∞ = = lim→∞
and there exists an integer such that ≤ ≤ for all then
lim
→∞=
Example 174 Show that the sequence {} =
©
(−1) 1!ª converges, and find its limit.
Solve: Use, for ≥ 4
! = 1· 2 · 3 · 4 · 5 · 6 · · = 24 · 5 · 6 · · and
2= 2· 2 · 2 · 2 · 2 · 2 · 2 = 16 · 2 · 2 · · 2 This implies that ≥ 4 2 ! We have
−1 2 ≤ (−1) 1 ! ≤ 1 2 for ≥ 4
Theorem 79 For the sequence {} if
lim
→∞|| = 0 then lim→∞ = 0
Monotonic Sequences and Bounded Sequences:
Definition 38 A sequence {} is monotonic if terms are nondecreasing
1 ≤ 2 ≤ 3 ≤ · · · ≤ ≤ · · ·
or if its terms are nonincreasing
1 ≥ 2 ≥ 3 ≥ · · · ≥ ≥ · · ·
Example 175 Determine whether each sequence having the given th term is monotonic.
() = 3 + (−1) () = 1+2 (yes) () =
2
2−1 (1 2 3)
Definition 39 1. A sequence {} is bounded above if there is a real number
such that ≤ for all The number is called an upper bound of
the sequence.
2. A sequence {} is bounded below if there is a real number such that
≥ for all The number is called an lower bound of the sequence.
3. A sequence {} is bounded if it is bounded above and bounded below.
Theorem 80 If a sequence {} is bounded and monotonic, then it
con-verges. Example 176 () {} = ©1 ª () {} = n 2 +1 o
(monot. but not bounded) () {} = {(−1)} (bounded, but not monot.)
