Lesson 6.pdf - 6 Convergence of Sequences Sequences...

6. Convergence of Sequences
July 15, 2019
Sequences A sequence is a list of real numbers with a particular order. We denote a
sequence as
( s
n
) = ( s
1
, s
2
, s
3
, ..., s
n
, ... ) .
Each s
n
is called a term of this sequence. 1
[Example] There are several ways to describe a sequence as follows.
1. Given the formula for all the terms, write down the sequence.
(a) s
n
=
1
n
. This is the sequence
( s
n
) = 1 ,
1
2
,
1
3
, ... .
(b) s
n
= n - 1
n
. This is the sequence
( s
n
) = 0 ,
1
2
,
2
3
,
3
4
, ... .
(c) s
n
= ( - 1)
n
n
2
. This is the sequence

2. By giving the first few terms to establish a pattern, leaving it to you to find the
function.
(a) ( s
n
) = (0 , 1 , 0 , 1 , 0 , 1 , .... ). The pattern here is obvious. We can write it as
(b) ( s
n
) = 2 ,
5
2
,
3
,
4
,
2
, ... . What is s
n
=?.
s
n
=
n
2
+ 1
n
.
(c) ( s
n
) = (2 , 4 , 8 , 16 , 32 , .... ). What is s
6
? You can find
3. By a recursion formula.
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