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
( s
n
) =
(
- 1 , 4 , - 9 , 16 , ..., ( - 1)
2
n
2
, ...
)
(d)
s
n
=
(
1
n
, n odd.
( - 1)
n/ 2
n
2
, n even.
This is the sequence
( s
n
) = 1 , - 4 ,
1
3
, 16 ,
1
5
, - 36 ,
1
7
, 64 , ...
1
It is also denoted by ( s
n
)
∞
n =1
, or { s
n
} , or { s
n
}
∞
n =1
.
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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
s
n
=
(
0 , n odd,
1 , n even.
or we can write s
n
=
1 - ( - 1)
n
2
. The second expression is better.
(b) ( s
n
) = 2 ,
5
2
,
10
3
,
17
4
,
26
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
s
n
= 2
n
= ⇒ s
6
= 64 .
3. By a recursion formula.
(a) s
n +1
=
1
n +1
s
n
and s
1
= 1. The first 5 terms are
1 ,
1
2
,
1
6
,
1
24
,
1
120
, ...
Assuming that the pattern continues:
s
n
=
1
n !
, ∀ n ∈ N .
In fact we can prove it: s
n +1
=
1
n +1
s
n
=
1
n +1
·
1
n
s
n - 1
= ...... =
1
n +1
·
1
n
· ... ·
1
2
s
1
=
1
( n +1)!
.
(b) s
n +1
=
1
2
( s
n
+ 1), s
1
= 1. The first 5 terms are
(1 , 1 , 1 , 1 , 1 , ... ) .
Assuming that the pattern continues
s
n
= 1 , ∀ n ∈ N .
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In fact, we can prove it.
s
n +1
=
1
2
( s
n
+ 1) =
1
2
s
n
+
1
2
=
1
2
·
1
2
( s
n - 1
+ 1) +
1
2
=
1
2
2
s
n - 1
+
1
2
+
1
2
2
=
1
2
2
· (
1
2
s
n - 2
+ 1) +
1
2
+
1
2
2
=
1
2
3
s
n - 2
+
1
2
+
1
2
2
+
1
2
3
= ......
=
1
2
n - 1
s
1
+
1
2
+
1
2
2
+
1
2
3
+ ... +
1
2
n - 1
=
1
2
n - 1
+
1
2
(1 -
1
2 n - 1
)
1 -
1
2
=
1
2
n - 1
+ 1 -
1
2
n - 1
= 1 .
More simply, we may use mathematical induction to prove it. Let S be the set
of positive integers for which this formula holds. We see 1 ∈ S . Assume k ∈ S ,
i.e., s
k
= 1. Then s
k +1
=
1
2
( s
k
+ 1) =
1
2
· 2 = 1 so that k + 1 ∈ S . We are done.
The induction method is better.
Convergence of a sequence Definition
1. A sequence ( s
n
) converges to the number s if for any δ > 0, there corresponds a positive
integer N such that
| s
n
- s | < for all n > N. (1)
The number s is called the limit of this sequence.
2. “( s
n
) converges to s ” is denoted by
lim
n →∞
s
n
= s, or lim s
n
= s, or s
n
→ s.
3. A sequence that does not converge is said to diverge .
[Examples and remarks]
1. After Newton and Leibniz, people did not know how to define “limit”. The first
definition of limit was given by Augustin Louis Cauchy (1789-1857) without - N
language:
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When the values successively assigned to the same variable indefinitely