Chapter 1: SEQUENCE AND SERIES
Content
Definition. Convergent, divergent, bounded & monotone sequences. Infinite sums: Basics, Convergence and divergence, Tests for convergence for positive term series, Alternating series – Leibnitz test, Absolute convergence, conditional convergence, Convergence of power series.
1.1. Definition: A
sequence is a list of numbers 
in a given order.
Each of 
and so on
represents a number.These are the terms of the sequence.
For example, the sequence 2, 4, 6, 8, 10, 12,…., 2n,….
has first term 
second term 
and nth term 
Order is important.
The sequence 2, 4, 6, 8 . . . is not the same as the sequence 4, 2, 6, 8 . . . .
The integer n is
called the index of 
, and indicates where occurs in the list.
Examples
1.2. Convergence and Divergence of Sequence
Definition: A sequence which has a finite limit is called a convergent sequence.
If 
, we say that 
converges to l.
Definition: The
sequence 
converges to
the number L if for every positive number 
there
corresponds an integer N such that for all n,
If no such number L exists, we say that diverges.
If converges to L,
we write 
, or simply 
, and
call L the limit of the sequence
Consider the sequence when 
. We find the values of for 
When 
When 
When 
When 
When 
When 
Thus, we see that as n becomes larger, 
becomes smaller.
In this case we can conclude that the sequence is convergent and converges to 1 or 
as 
i.e. 
Example 1: Discuss
the convergence of the sequences 
where
Solution: We have given
Putting 
in (1), we get,
, 
,
, 
…………. ….. …….. ………. ………………….
Although the first term is infinity, we have to consider large number of term.
Here we observe that
is less than 1 if n is even and is greater than 1 if n is odd.
Thus, when n is even, take 
and when n is odd, take 
Further as 
, taking limit in (2) and
(3), we get
Also,
Thus we get as 
and also 
.
Hence, 
is convergent and the limit is 1.
Graphically the convergence of series can be seen as follows:
1.3. Divergence of a sequence:
• Definition: A sequence which does not converge is called a divergent sequence.
• There are three types of divergent sequence.
(i) A sequence diverging to +∞.
(ii) A sequence diverging to -∞.
(iii) A sequence which oscillates.
Example 2: Prove that the sequence 
does not
converge.
Solution: we have given
Therefore, the sequence has two limit points, hence does not converge.
Example 3: Prove that the sequence 
converges. Hence find the natural number n.
Solution: we have given
As
approaches to infinity 
.
