Module 1
:
Real Numbers, Functions and Sequences
Lectur
e
3
:
Monotone Sequence and Limit theorem [ Section 3.2 : Limit Theorems on Sequences ]
(i)
Expand
by binomial theorem and use the fact
to show that
is bounded.
(ii)
Using the fact that for all
,
show that
is monotonically increasing.
(iii)
Use completeness property to deduce
exists.
The limit, denoted by
, is called
Euler's number
. It is an irrational number and the above sequence is used to find its approximate values.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
by binomial theorem and use the fact 
to show that 
, 
exists.
, is called Euler's number. It is an irrational number and the above sequence is used to find its approximate values. 
