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Math 117 – May 12, 2011
12 Convergence of sequences
A
sequence
is a function whose domain is the set of natural numbers
N
. We usually denote the values
of the function by
s
n
, instead of
s
(
n
), and call these values the
terms of the sequence
. We will be
primarily interested in sequences of real numbers, in which case the terms of the sequence
s
n
∈
R
for
every
n
∈
N
. We use the notation (
s
n
) for the entire sequence.
Deﬁnition 12.1.
A sequence (
s
n
) is said to converge to
s
∈
R
, if for every
± >
0 there exists
n
∈
N
,
such that for every
n > N
,

s
n

s

< ±
.
The number
s
is called the limit of the sequence, and is denoted by
s
= lim
s
n
. If the limit exists,
(
s
n
) is called a convergent sequence. If a sequence does not converge to any real number, it is said to
be divergent.
When trying to show that the real number
s
is the limit of the sequence (
s
n
), it is sometimes more
convenient to consider the sequence made of the diﬀerences (
s
n

s
). The following theorem illustrates
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This note was uploaded on 05/31/2011 for the course MATH 117 taught by Professor Akhmedov,a during the Spring '08 term at UCSB.
 Spring '08
 Akhmedov,A
 Natural Numbers

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