CHAPTER1 MAT455

Pensyarah:Sumarni Abu Bakar; FSKM, Tingkat 3-52; 0126163540(sms)

1. SEQUENCES AND SERIES

1.0 Introduction to sequences

 A sequence can be defined as an unending succession of numbers

 The numbers in a sequence are called the term of the sequence
 A sequence can be represented by a formula and can be written in bracket
notation,

Eg: {𝑎𝑛 }𝑛=∞

𝑛=1 = {𝑎1 , 𝑎2 , 𝑎3 , … … … … }

Sequence can be classified into 2 groups:

1) Finite sequence : a sequence that contains finite number of terms.

Eg: {𝑎𝑛 }𝑛=𝑘

𝑛=1 = a1 , a2 , a3 , ………, ak

2) Infinite sequence : a sequence with infinite number of terms.

Eg: {𝑎𝑛 }𝑛=∞

𝑛=1 = a1 , a2 , a3 , ………

Definition 1

1) A sequence {𝑎𝑛 }𝑛=∞

𝑛=1 is said to converge iff

lim an = L
n→∞

2) A sequence that does not converge to some finite number is said to be

divergent

lim an = ∞
n→∞

Theorem 1
If lim 𝑓(𝑥) = 𝐿 and f(n) = an when n is an integer, then lim 𝑎𝑛 = 𝐿.
𝑥→∞ 𝑛→∞

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Example 1
Determine whether the following sequences converge or diverge.

𝑛 ∞
a) {𝑛+1}
𝑛=1

∞
𝑛2
b) {2𝑛−1}
𝑛=1

ln 𝑛 ∞
c) { }
𝑛 𝑛=1

NOTE:
0 −1 < 𝑟 < 1
𝑥
1. lim 𝑟 = { 1 𝑟=1
𝑥→∞
∞ |𝑟| > 1
𝑥+𝑘 𝑥
2. lim ( ) = 𝑒𝑘
𝑥→∞ 𝑥

𝑥 𝑥
3. lim (𝑥+𝑘) = 𝑒 −𝑘
𝑥→∞

Definition 2
A sequence {𝑎𝑛 } is called increasing if an < an+1 for all n ≥ 1, that is a1 < a2 < a3 <….
It is called decreasing if an > an+1 for all n ≥ 1. It is called monotonic if it is either
increasing or decreasing

Example 2
Show that each of the following sequence is decreasing
3 𝑛
a) {𝑛+5} b) {𝑛2 +1}

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1.1 Infinite series

If we add the terms of an infinite sequence {𝑎𝑛 }𝑛=∞

𝑛=1 we will get an expression of the
form
a1 + a2 + a3 + …………

which is called an infinite series (or just series) and is denoted by

∑𝑛=∞
𝑛=1 𝑎𝑛

Definition 3
An infinite series is the summation of terms in an infinite sequence.

If Sn is the summation of n terms, then

S1 = a 1
S 2 = a 1 + a2
S3 = a1 + a2 + a3 etc

The number Sn is called the nth partial sum of the series and {Sn} is the sequence of
partial sum.

Definition 4
𝑛=1 𝑎𝑛 = a1 + a2 + a3 + …………, and let Sn denote its nth partial sum,
Given a series ∑𝑛=∞

Sn = ∑𝑛𝑖=1 𝑎𝑖 = a1 + a2 + a3 + …………+ an

If the sequence of the partial sum {Sn} is convergent and lim 𝑆𝑛 = 𝑠 exist as a real
𝑛→∞
number, then the series ∑𝑛=∞
𝑛=1 𝑎𝑛 is called convergent and we write

a1 + a2 + a3 + …………= s or ∑𝑛=∞
𝑛=1 𝑎𝑛 = s

The number s is called the sum of the series. Otherwise the series is called divergent.

Thus the sum of a series is the limit of the sequence of the partial sums. So when we write ∑𝑛=∞
𝑛=1 𝑎𝑛 = s, we mean that by
adding sufficiently many terms of the series, we can get as close as we like to the number s.

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1.2 Types of series

a) The Geometric series:

∑∞
𝑛=1 𝑎𝑟
𝑛−1
= a + ar + ar2 + ar3 + ….
𝑎
is convergent if |𝑟| < 1 and its sum is ∑∞
𝑛=1 𝑎𝑟
𝑛−1
= 1−𝑟 .

If |𝑟| > 1, the geometric series is divergent.

Example 3
Determine whether the following series is convergent or divergent
3 2𝑛 3 𝑛−1
a) ∑∞
𝑛=1 2𝑛 b) ∑∞
𝑛=1 100 c) ∑∞
𝑛=1 2 (4)

b) The Harmonic series:

1 1 1
∑∞
𝑛=1 𝑛 = 1 + +3 +⋯ is a divergent series
2

c) The p-series (also known as hyperharmonic series)

1 1 1
∑∞
𝑛=1 =1+ + + … . . , where p > 0
𝑛𝑝 2𝑝 3𝑝

i) converges if p > 1

ii) diverges if 0 < p ≤ 1

Example 4
Determine if the following series is convergent or divergent
1 1 2 1
a) ∑∞
𝑛=1 𝑛3 b) ∑∞𝑛=1 3 c) ∑∞
𝑛=1 (3𝑛 + 𝑛)
𝑛 √𝑛 √

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d) Telescoping Series
𝑎
Series of the form ∑∞
𝑛=1 𝑓(𝑛)𝑔(𝑛) known as the telescoping series can be written in the
𝐴 𝐵
form of ∑∞
𝑛=1(𝑓(𝑛) + 𝑔(𝑛)) using the method of partial fraction.

Example 5
1
Show that the series ∑∞
𝑛=1 𝑛 (𝑛+1) is convergent and find its sum.

Example 6
3
Show that the series ∑∞
𝑛=1 (𝑛+3) (𝑛+4) is convergent and find its sum.

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