Chapter 4
Sequences and Series
1. Sequences
2. Series
3. Special Sequences and Series
Section 1
Sequences
Sequences
• A sequence 𝑎𝑛 𝑛≥1 is as a list of numbers written in order:
𝑎1 , 𝑎2 , 𝑎3 , 𝑎4 , ⋯ , 𝑎𝑛 , ⋯
• Sequences can be represented as points on the real line or as
points in the plane where the horizontal axis n is the index
number of the term 𝑎𝑛 and the vertical axis is its value.
• The general term of a sequence, or the 𝑛𝑡ℎ term and written
as 𝑎𝑛 is an algebraic expression that relates the term to its
position number in the sequence.
Representing a sequence
General Term of a Sequence
Example: Write the first four terms of each sequence:
1. 𝑎𝑛 = 𝑛 + 2 3, 4, 5, 6
2. 𝑎𝑛 = 3𝑛 − 2 1,4,7,10
−1 𝑛 1 1 1
3. 𝑎𝑛 = 𝑛
−1, , − ,
2 3 4
4. 𝑎𝑛 = −1 𝑛−1 5𝑛 5, – 25, 125, – 625
Limit of a Sequence
• A sequence 𝑎𝑛 has the limit 𝐿 if we can make the terms
𝑎𝑛 as close as we like by taking 𝑛 sufficiently large.
• We write lim 𝑎𝑛 = 𝐿 or 𝑎𝑛 → 𝐿 𝑎𝑠 𝑛 → ∞
𝑛→∞
𝑦 = 𝐿 is a horizontal asymptote
Limit of a Sequence
• If lim 𝑎𝑛 exists, we say that the sequence converges.
𝑛→∞
Note that a sequence converges if it approaches a finite limit.
• If the sequence does not converge, we will say that it diverges.
Note that a sequence diverges if it approaches infinity or if it
does not approach anything.
Limit of a Sequence
convergent convergent
divergent divergent divergent
Limit of a Sequence
𝑎𝑖 𝑛𝑖 +⋯
Consider a sequence of the form
𝑎𝑗 𝑛𝑗 +⋯
1. First case: 𝑖 < 𝑗. The sequence converges to 0.
𝑎𝑖
2. Second case: 𝑖 = 𝑗. The sequence converges to
𝑎𝑗
3. Third case: 𝑖 > 𝑗. The sequence diverges to ±∞
The limit laws
If 𝑎𝑛 and 𝑏𝑛 are convergent sequences with
lim 𝑎𝑛 = 𝐴 and lim 𝑏𝑛 = 𝐵.
𝑛→∞ 𝑛→∞
• Sum Rule: lim 𝑎𝑛 + 𝑏𝑛 = 𝐴 + 𝐵
𝑛→∞
• Difference Rule: lim 𝑎𝑛 − 𝑏𝑛 = 𝐴 − 𝐵
𝑛→∞
• Constant Rule: lim 𝑘 ⋅ 𝑎𝑛 = 𝑘 ⋅ 𝐴
𝑛→∞
• Product Rule: lim 𝑎𝑛 ⋅ 𝑏𝑛 = 𝐴 ⋅ 𝐵
𝑛→∞
𝑎𝑛 𝐴
• Quotient Rule: lim =
𝑛→∞ 𝑏𝑛 𝐵
• Power Rule: lim 𝑎𝑛 𝑝 = 𝐴𝑝
𝑛→∞
Section 2
Series
Series
• A series is the sum of a sequence.
• Series are represented using the summation notation
• Finite series
𝑛
𝑎𝑘 = 𝑎1 + 𝑎2 + 𝑎3 + ⋯ + 𝑎𝑛
𝑘=1
• Infinite series
∞
𝑎𝑛 = 𝑎1 + 𝑎2 + 𝑎3 + ⋯
𝑛=1
Summation Notation
Example:
6
1. 𝑘 2 = 12 + 22 + 32 + 42 + 52 + 62
𝑘=1
= 1 + 4 + 9 + 16 + 25 + 36
5
𝑘 1 2 3 4 5
2. 2 = 2 + 2 + 2 + 2 + 2
𝑘 +1 1 +1 2 +1 3 +1 4 +1 5 +1
𝑘=1
1 2 3 4 5
= + + + +
2 5 10 17 26
6
−1 𝑘+1 1 1 1 1 1 1
3. = − + − + −
2𝑘 2 4 6 8 10 12
𝑘=1
Convergence/Divergence
∞
We say that an infinite series converges if the sum 𝑎𝑛
𝑛=1
is finite, otherwise we will say that it diverges.
Example:
∞ 𝑛
1
1. converges.
2
𝑛=1
∞
2. 2𝑛 diverges.
𝑛=1
Laws of Series
∞ ∞
If 𝑎𝑛 and 𝑏𝑛 both converge, then
𝑛=1 𝑛=1
∞ ∞ ∞
1. 𝑎𝑛 ± 𝑏𝑛 = 𝑎𝑛 ± 𝑏𝑛
𝑛=1 𝑛=1 𝑛=1
∞ ∞
2. 𝑘 ⋅ 𝑎𝑛 = 𝑘 ⋅ 𝑎𝑛
𝑛=1 𝑛=1
∞ ∞ ∞
𝑎𝑛 ⋅ 𝑏𝑛 ≠ 𝑎𝑛 ⋅ 𝑏𝑛
𝑛=1 𝑛=1 𝑛=1
Section 3
Special Sequences and Series
Arithmetic Sequence
• A sequence 𝑎𝑛 is called an arithmetic sequence if there
is a constant 𝑑 such that
𝑎𝑛+1 = 𝑎𝑛 + 𝑑
• The 𝑛𝑡ℎ term can be calculated using the formula
𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
Example:
𝑎1 = 3, 𝑎2 = 6, 𝑎3 = 9, 𝑎4 = 12, ⋯ calculate 𝑎40
𝑑=3
𝑎40 = 𝑎1 + 40 − 1 𝑑 = 3 + 39 ⋅ 3 = 120
Arithmetic Series
• A series is called an arithmetic series if the underlying
sequence is an arithmetic sequence.
• The sum of a finite arithmetic series can be calculated
using the formula
𝑛
𝑛
𝑎𝑘 = 𝑎1 + 𝑎𝑛
2
𝑘=1
• An infinite arithmetic series diverges.
Arithmetic Series
Example:
Calculate the sum of the first 30 terms of the sequence
𝑎1 = 3, 𝑎2 = 8, 𝑎3 = 13, 𝑎4 = 18, ⋯
𝑑=5
𝑎30 = 𝑎1 + 30 − 1 𝑑 = 3 + 29 ⋅ 5 = 148
30
30 30
𝑎𝑛 = 𝑎1 + 𝑎30 = 3 + 148 = 2,265
2 2
𝑛=1
Geometric Sequence
• A sequence 𝑎𝑛 is called a geometric sequence if there
is a constant 𝑟 > 0 such that
𝑎𝑛+1 = 𝑟 ∙ 𝑎𝑛
• The 𝑛𝑡ℎ term can be calculated using the formula
𝑎𝑛 = 𝑎1 𝑟 𝑛−1
Example:
𝑎1 = 6, 𝑎2 = 12, 𝑎3 = 24, 𝑎4 = 48,… calculate 𝑎10
𝑟=2
𝑎10 = 𝑎1 𝑟 9 = 6 ⋅ 29 = 3072
Finite Geometric Series
• A series is called a geometric series if the underlying
sequence is a geometric sequence.
• If 𝑟 ≠ 1, the sum of a finite geometric series can be
calculated using the formula
𝑛
1 − 𝑟𝑛
𝑎𝑘 = 𝑎1
1−𝑟
𝑘=1
• If 𝑟 = 1, a geometric sequence is constant, and the sum of
a finite geometric series is just 𝑛𝑎1 .
Infinite Geometric Series
• If 0 < 𝑟 < 1, an infinite geometric series converges, and
its sum can be calculated using the formula
∞
𝑎1
𝑎𝑛 =
1−𝑟
𝑛=1
• If 𝑟 ≥ 1, an infinite geometric series diverges.
Geometric Series
Example:
1
Consider the sequence 𝑎1 = 4, 𝑎2 = 2, 𝑎3 = 1, 𝑎4 = ,⋯
2
1. Calculate the sum of the first 8 terms.
1
𝑟=
2
8
8 1
1 − 𝑟8 1−
𝑎𝑛 = 𝑎1 =4 2 = 7.96875
1−𝑟 1
𝑛=1 1−
2
2. Calculate the sum of the infinite series.
∞
𝑎1 4
𝑎𝑛 = = =8
1−𝑟 1− 1
𝑛=1
2
Geometric Series
Example:
Consider the sequence 𝑎1 = 2, 𝑎2 = 6, 𝑎3 = 18, 𝑎4 = 54, ⋯
1. Calculate the sum of the first 12 terms.
𝑟=3
12
1 − 𝑟12 1 − 312
𝑎𝑛 = 𝑎1 =2 = 531,440
1−𝑟 1−3
𝑛=1
2. Calculate the sum of the infinite series.
The infinite series diverges.
