Course Material 2.1 Sequences and Their Convergence
Introduction
By a sequence, we mean terms coming one after another just as in the frames of a
movie. We are concerned about real sequences or real numbers coming one after another.
This may be represented as 𝑥1 , 𝑥2 , 𝑥3 , … … … … , 𝑥𝑛 −1 , 𝑥𝑛 , 𝑥𝑛+1 , … … …where each 𝑥𝑛 are real
numbers and 𝑥𝑛 is said to be the 𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 of the sequence. This sequence may also be
represented as 𝑥𝑛 or some times 𝑥𝑛 . For a formal definition of sequences, we consider a
sequence as a function defined from the set of positive integers with the idea that as 𝑛
increases from 1, the sequence will run from the first term 𝑥1 , then the second term 𝑥2 and
so on.
Definition – Sequences in a set 𝑿
Let 𝑋 be a non empty set. Then a sequence in 𝑋 is a function 𝑓: ℕ → 𝑋. We let 𝑓 𝑛 = 𝑥𝑛 for
each 𝑛 ∈ ℕ and is called the 𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 of the sequence.
Note:
With this definition we can consider sequences of real numbers, sequences of
complex numbers, sequences of functions, sequences of vectors etc.
Definition - Real Sequences
A real sequence or a sequence of real numbers is a function 𝑓: ℕ → ℝ. We let 𝑓 𝑛 = 𝑥𝑛 for
each 𝑛 ∈ ℕ and is called the 𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 of the sequence.
Examples
1. Let 𝑥𝑛 = 𝑐 for all 𝑛 ∈ ℕ , where 𝑐 is a constant. Then the sequence 𝑐 is
𝑐, 𝑐, 𝑐, 𝑐 … ..,and is called a constant sequence.
2. Let 𝑥𝑛 = 𝑛 for all 𝑛 ∈ ℕ then 𝑥𝑛 is 1, 2, 3, 4, … 𝑛, 𝑛 + 1, … ..
1 1 1 1 1 1
3. Let 𝑥𝑛 = 𝑛 for all 𝑛 ∈ ℕ then 𝑥𝑛 is 1, 2 , 3 , 4 , … . . 𝑛 , 𝑛+1 , … … ..
1 1 1 1 1 1
4. Let 𝑥𝑛 = 2𝑛 for all 𝑛 ∈ ℕ then 𝑥𝑛 is 1, 2 , 22 , 23 , … . . 2𝑛 , 2𝑛 +1 , … … ..
Introduction
By a sequence, we mean terms coming one after another just as in the frames of a
movie. We are concerned about real sequences or real numbers coming one after another.
This may be represented as 𝑥1 , 𝑥2 , 𝑥3 , … … … … , 𝑥𝑛 −1 , 𝑥𝑛 , 𝑥𝑛+1 , … … …where each 𝑥𝑛 are real
numbers and 𝑥𝑛 is said to be the 𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 of the sequence. This sequence may also be
represented as 𝑥𝑛 or some times 𝑥𝑛 . For a formal definition of sequences, we consider a
sequence as a function defined from the set of positive integers with the idea that as 𝑛
increases from 1, the sequence will run from the first term 𝑥1 , then the second term 𝑥2 and
so on.
Definition – Sequences in a set 𝑿
Let 𝑋 be a non empty set. Then a sequence in 𝑋 is a function 𝑓: ℕ → 𝑋. We let 𝑓 𝑛 = 𝑥𝑛 for
each 𝑛 ∈ ℕ and is called the 𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 of the sequence.
Note:
With this definition we can consider sequences of real numbers, sequences of
complex numbers, sequences of functions, sequences of vectors etc.
Definition - Real Sequences
A real sequence or a sequence of real numbers is a function 𝑓: ℕ → ℝ. We let 𝑓 𝑛 = 𝑥𝑛 for
each 𝑛 ∈ ℕ and is called the 𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 of the sequence.
Examples
1. Let 𝑥𝑛 = 𝑐 for all 𝑛 ∈ ℕ , where 𝑐 is a constant. Then the sequence 𝑐 is
𝑐, 𝑐, 𝑐, 𝑐 … ..,and is called a constant sequence.
2. Let 𝑥𝑛 = 𝑛 for all 𝑛 ∈ ℕ then 𝑥𝑛 is 1, 2, 3, 4, … 𝑛, 𝑛 + 1, … ..
1 1 1 1 1 1
3. Let 𝑥𝑛 = 𝑛 for all 𝑛 ∈ ℕ then 𝑥𝑛 is 1, 2 , 3 , 4 , … . . 𝑛 , 𝑛+1 , … … ..
1 1 1 1 1 1
4. Let 𝑥𝑛 = 2𝑛 for all 𝑛 ∈ ℕ then 𝑥𝑛 is 1, 2 , 22 , 23 , … . . 2𝑛 , 2𝑛 +1 , … … ..
