Chapter 10: Infinite Sequences and Series
Section 10.1: Sequences
Definition: A sequence is an ordered list of numbers
a1 , a2 , a3 , . . . , an , . . . .
The number an is called the nth term of the sequence.
Notation: Sequences are typically denoted by {an }∞
n=1 or {an }.
Example: Write out the first 5 terms of the sequence
n
n+1
∞
.
n=1
The first 5 terms of this sequence are
1 2 3 4 5
, , , , .
2 3 4 5 6
Example: Find a formula for the general term an of the following sequences:
1 1
1 1 1
,− , ,− , ,...
(a)
2 4 8 16 32
The general term of the sequence is an =
(b)
(−1)n+1
for n ≥ 1.
2n
1 2 3 4 5
, , , , ,...
4 9 16 25 36
The general term of the sequence is an =
n
for n ≥ 1.
(n + 1)2
(c) {−2, 2, −2, 2, −2, 2, . . .}
The general term of the sequence is an = (−2)n for n ≥ 1.
Definition: If lim an = L, then we say the sequence converges with limit L. Otherwise,
n→∞
we say the sequence diverges.
Example: Determine whether the following sequences converge or diverge:
(a) an =
4n − 3
3n + 4
The sequence converges with limit
lim
n→∞
r
(b) an =
4
4n − 3
= .
3n + 4
3
2n2 + 3
18n2 − 6
The sequence converges with limit
r
r
r
2n2 + 3
2
1
1
lim
=
=
= .
2
n→∞
18n − 6
18
9
3
(c) an =
n
ln n
By L’Hopital’s Rule,
n
1
= lim
= lim n = ∞.
n→∞ ln n
n→∞ 1
n→∞
n
The sequence diverges to ∞.
lim
(d) an = ln(4n + 1) − ln(2n + 3)
The sequence converges with limit
lim [ln(4n + 1) − ln(2n + 3)] = lim ln
n→∞
n→∞
4n + 1
2n + 3
= ln(2).
(e) an =
sin n
n2
By the Squeeze Theorem, the sequence converges with limit 0. Indeed,
1
sin n
1
≤ lim
≤ lim 2 = 0.
2
2
n→∞ n
n→∞ n
n→∞ n
0 = − lim
(f) {1, 0, 1, 0, 1, 0, . . .}
The sequence diverges by oscillation. In particular, the limit lim an does not exist.
n→∞
(g) an =
√
n
n
Let L = lim
n→∞
√
n
n. Then
ln(L) =
lim ln(n1/n )
n→∞
1
ln(n)
n
1
= lim
n→∞ n
= 0.
=
lim
n→∞
Thus, L = e0 = 1. The sequence converges with limit 1.
Definition: A sequence {an } is bounded above if there exists a number M such that an ≤ M
for all n ≥ 1 and bounded below if there exists a number N such that N ≤ an for all
n ≥ 1. A sequence is called bounded if it is bounded above and below. If a sequence is not
bounded, it is called unbounded.
Example: Determine whether the following sequences are bounded or unbounded.
∞
1
(a)
n n=1
The sequence is bounded since
0<
1
≤ 1 for n ≥ 1.
n
(b)
2n3 + 6n
n2 + 1
∞
n=1
The sequence is unbounded since
2n3 + 6n
= ∞.
n→∞ n2 + 1
lim
(c) {e−n }∞
n=1
The sequence is bounded since
0 < e−n =
1
≤ 1 for n ≥ 1.
en
Definition: A sequence {an } is called increasing if an < an+1 for all n ≥ 1 and decreasing
if an > an+1 for all n ≥ 1. A sequence is called monotonic or monotone if it is either
increasing or decreasing.
Example: Determine whether the following sequences are increasing, decreasing, or not monotone.
n
1
(a) an =
3
The sequence is decreasing since
an+1 =
(b) an = 1 −
1
3n+1
<
1
= an for n ≥ 1.
3n
1
n2
The sequence is increasing since
an+1 = 1 −
1
1
≥ 1 − 2 = an .
2
(n + 1)
n
n
2
(c) −
3
The sequence is not monotone.
Theorem: (Monotone Convergence Theorem)
Every bounded, monotone sequence is convergent.
3
for
an
n ≥ 1. Given that the sequence is increasing and bounded, find the limit of the sequence.
Example: Consider the sequence whose terms are defined by a1 = 2 and an+1 = 4 −
The sequence converges by the Monotone Convergence Theorem. Let
L = lim an .
n→∞
By definition of the sequence,
lim an+1 =
n→∞
L =
L2
L2 − 4L + 3
(L − 1)(L − 3)
L
=
=
=
=
3
lim 4 −
n→∞
an
3
4−
L
4L − 3
0
0
1, 3.
Since a1 = 2 > 1 and {an } is increasing, L 6= 1. Thus, the limit of the sequence if 3.