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Sequences

# Sequences - ECE 3250 MORE ON REAL AND COMPLEX NUMBERS Fall...

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ECE 3250 MORE ON REAL AND COMPLEX NUMBERS Fall 2008 My purpose in this handout is to summarize fairly briskly the main results on sequences and series of real and complex numbers. I’ll be lifting some of the definitions (e.g. con- vergent sequences, Cauchy sequences, etc.) from the NUMBERS handout, but I think it’s useful to collect the principal facts in one place in a kind of bulleted-list format. The lack of a lot of intervening text makes for some high-density mathematics, but the layout, I hope, will facilitate easy reference. To start with, I’ll assume you have some basic familiarity with the real numbers R and the complex numbers C . I’ll assume you understand the algebra of complex numbers at the level of the ALGEBRA OF COMPLEX NUMBERS handout. For a real number a , | a | denotes the absolute value of a ; for a complex number c , | c | denotes the magnitude of c . So if c = a + jb , with a and b in R , then | c | = p a 2 + b 2 . The distance between two real numbers a and b is the absolute value of a - b and the distance between two complex numbers c 1 and c 2 is the magnitude of c 1 - c 2 . To avoid having to type “real or complex numbers” a zillion times, I’ll use the notation F to denote the phrase “ R or C .” The “F” is supposed to mean “field.” Notational Convention: In what follows, F = R or C . Definition 1: A sequence in F is an ordered list of elements of F indexed by N . We use notation such as { a n } or { c n } to denote such a sequence. So, for example, { a n } = a 0 , a 1 , a 2 , a 3 , . . . . Definition 2: We say that a sequence { a n } in F converges to ¯ a F if and only if the distance between a n and ¯ a approaches zero as n → ∞ . In this case, we write lim n →∞ a n = ¯ a . A precise mathematical definition of convergence: { a n } converges to ¯ a if and only if for every > 0 there exists an integer N > 0 such that | a n - ¯ a | < for every n > N . Fact 1: A sequence { c n = a n + jb n } in C converges to ¯ c = ¯ a + j ¯ b C if and only if { a n } converges to ¯ a and { b n } converges to ¯ b . Proof: First of all, for every n N , | c n - ¯ c | = q | a n - ¯ a | 2 + | b n - ¯ b | 2 , If { c n } converges to ¯ c , then for every > 0 we can find N > 0 so that | c n - ¯ c | < when n > N . Hence for n > N , we have q | a n - ¯ a | 2 + | b n - ¯ b | 2 < , which implies that both | a n - ¯ a | < and | b n - ¯ b | < for every n > N . Accordingly, { a n } converges to ¯ a and { b n } converges to ¯ b . Conversely, if { a n } converges to ¯ a and { b n } converges to ¯ b , then for every > 0 we can find N > 0 so that both | a n - ¯ a | < 2 / 2 and | b n - ¯ b | < 2 / 2 when n > N . Hence for n > N , we have | c n - ¯ c | < p 2 / 2 + 2 / 2 = . 1

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2 Accordingly, { c n } converges to ¯ c . In words, Fact 1 states that a sequence of complex numbers converges if and only if the real-part sequence and imaginary-part sequence both converge, in which case the limit of the real parts is the real part of the limit, and the limit of the imaginary parts is the imaginary part of the limit.
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