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Mathematics

MATH 4389

Almus

Spring

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PART II. SEQUENCES OF REAL NUMBERS
II.1. CONVERGENCE
Deﬁnition 1. A sequence is a real-valued function f whose domain is the set positive integers
(N). The numbers f(1),f (2), ··· are called the terms of the sequence.
Notation Function notation vs subscript notation:
f(1) ≡ 1 ,f (2) ≡2s ,··· ,f (n) n s , ··· .
In discussing sequences the subscript notation is much more common than functional notation. We’ll
use subscript notation throughout our treatment of analysis.
Specifying a sequence There are several ways to specify a sequence.
1. By giving the function. For example:
1 1 1 1 1 1
(a) sn= n or {sn} = n . This is the sequence {12 ,3 ,4 ,...,n,... }.
(b) s = n − 1. This is the sequence {0, ,2 , 3,...,n − 1,... }.
n n 2 3 4 n
(c) sn=( −1) n . This is the sequence {−1,4,−9,16,..., (−1) n ,... }.
2. By giving the ﬁrst few terms to establish a pattern, leaving it to you to ﬁnd the function. This
is risky – it might not be easy to recognize the pattern and/or you can be misled.
(a) {sn} = {0,1,0,1,0,1,... }( The pattern here is obvious; can you devise the function? It’s
1 − (−1)n) 0, n odd
sn= or s n=
2 1, n even
2
(b) {s } = 2, 5,10,17 ,6 ,... ,s = n +1 .
n 2 3 4 5 n n
(c) {sn} = {2,4,8,16,32,... }. What is6s ? What is the function? While you might say 64
n
and s n2 , the function I have in mind gives s6= π/6:
n π 64
sn=2 +( n − 1)(n − 2)(n − 3)(n − 4)(n − 5) −
720 120
3. By a recursion formula. For example:
(a) s = 1 s ,s = 1. The ﬁrst 5 terms are 1,1, , 1 , 1 ,.... Assuming that
n+1 n +1 n 1 2 6 24 120
1
the pattern continues n = .
n!
1
(b) sn+1 = (n +1) ,s 1 = 1. The ﬁrst 5 terms are {1,1,1,1,1,... }. Assuming that the
2
pattern continues n = 1 for all n; {n } is a “constant” sequence.
13 Deﬁnition 2. A sequence {s }nconverges to the number s if to each > 0 there corresponds
a positive integer N such that
|s − s| for all n>N.
n
The number s is called the limit of the sequence.
Notation “{s } converges to s” is denoted by
n
n→∞ sn= s, or by limsn= s, or by n → s.
A sequence that does not converge is said to diverge.
Examples Which of the sequences given above converge and which diverge; give the limits of the
convergent sequences.
THEOREM 1. If s → s and s → t, then s = t. That is, the limit of a convergent sequence
is unique.
Proof: Suppose s 6= t. Assume t>s and let = t − s. Since sn→ s, there exists a positive
integer N such that |s − s | / 2 for all n>N . Since s → t, there exists a positive integer
1 n 1 n
N 2 such that |t−s n / 2 for all n>N 2. Let N = max{N ,N1} a2d choose a positive integer
k>N . Then
t − s = |t − s| = |tk− s k s − s|≤| t k s | + |sk− s |+< = = t − s,
2 2
a contradiction. Therefore, s = t.
THEOREM 2. If {s } connerges, then {s } is bnunded.
Proof: Suppose s → s. There exists a positive integer N such that |s−s | < 1 for all n>N .
n n
Therefore, it follows that
|sn| = |n − s + s|≤| sn− s| + |s| < 1+ |s| for alln>N.
Let M = max{|s |, |s |, ..., |s |, 1+ |s|}. Then |s | N.
If an→ 0, then s →n0.
Proof: Note ﬁrst that a ≥n0 for all n>N . Since a → 0,nthere exists a positive integer N 1
such that |an| < /k. Without loss of generality, assume that1N ≥ N. Then, for all n>N 1,
|sn− 0| = |sn|≤ ka n

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