Section 2.2: Sequences
Definition: A sequence is an ordered list of numbers
{an } = a0 , a1 , a2 , . . . , an , . . . .
The number an is called the nth term of the sequence. Sequences are typically denoted by
{an } or {an }∞
n=1 .
∞
n
Example: Write out the first 5 terms of the sequence
.
n + 1 n=1
Example: Find a formula for the general term an of the following sequences:
1 1 1
1 1
(a)
,− , ,− , ,...
2 4 8 16 32
(b)
1 2 3 4 5
, , , , ,...
4 9 16 25 36
(c) {−2, 2, −2, 2, −2, 2, . . .}
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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
r
(b) an =
(c) an =
2n2 + 3
18n2 − 6
n
ln n
(d) an = ln(4n + 1) − ln(2n + 3)
2
(e) an =
sin n
en
(f) {1, 0, 1, 0, 1, 0, . . .}
(g) an =
√
n
n
3
Note: Some sequences are defined recursively. For example,
an+1 = 2an ,
a0 = 1.
The terms of this sequence are
1, 2, 4, 8, 16, 32, 64, . . . .
Example: Consider the sequence defined by a0 = 2 and an+1 = 3an − 2. Find the first 5
terms of this sequence and determine whether the sequence converges.
Example: Find the limit of the sequence defined by a0 = 2 and an+1 = 4 −
4
3
.
an
Definition: A fixed point or equilibrium of a sequence {an } is a value such that an+1 = an .
Example: Find the fixed points of each sequence.
(a) an+1 =
an 4
+
3
3
(b) an+1 =
3
an − 2
5
Example: (The Fibonacci Sequence)
The growth of a rabbit population can be modeled by the recusive sequence
Nt+1 = Nt + Nt−1 ,
where N0 = N1 = 1.
(a) Find the first 7 terms of this sequence and determine whether the sequence converges.
(b) The golden ratio is defined as the limit
Nt+1
= φ.
t→∞ Nt
lim
Evaluate this limit to determine the golden ratio.
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