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**Unformatted text preview: **Numerical Sequences and Series Convergent Sequences 3.1 Definition A sequence { p n } in a metric space X is said to converge if there is a point p ∈ X with the following property: For every ǫ > 0 there is an integer N such that n ≥ N implies that d ( p n , p ) < ǫ . In this case we say that { p n } converges to p , or that p is the limit of { p n } , and we write p n → p , or lim n →∞ = p. If { p n } does not converge, it is said to diverge . 3.2 Theorem Let { p n } be a sequence in a metric space X . (a) { p n } converges to p ∈ X if and only if every neighborhood of P contains p n for all but finitely many n . (b) If p ∈ X , p ′ ∈ X , and if { p n } converges to p and to p ′ , then p ′ = p . (c) If { p n } converges, then { p n } is bounded. (d) If E ⊆ X and if p is a limit point of E , then there is sequence { p n } in E such that p = lim n →∞ p n . 3.3 Theorem Suppose { s n } , { t n } are complex sequences, and lim n →∞ s n = s , lim n →∞ t n = t . Then (a) lim n →∞ ( s n + t n ) = s + t ; (b) lim n →∞ cs n = cs, lim n →∞ ( c + s n ) = c + s for any number c ; (c) lim n →∞ s n t n = st ; (d) lim n →∞ 1 s n = 1 s , provided s n negationslash = 0 ( n = 1 , 2 , 3 ,... ) , and s negationslash = 0 . 3.4 Theorem (a) Suppose x n ∈ R k ( n = 1 , 2 , 3 ,... ) and x n = ( α 1 ,n,...,α k ,n ) . Then { x n } converges to x = ( α 1 ,...,α k ) if and only if lim n →∞ α j,n = α j (1 ≤ j ≤ k ) . (b) Suppose { x n } , { y n } are sequences in R k , { β n } is a sequence of real numbers, and x n → x , y n → y , β n → β . Then lim n →∞ ( x n + y n ) = x + y , lim n →∞ x n · y n = x · y , lim n →∞ β n x n = β x . Subsequences 3.5 Definition Given a sequence { p n } , consider a sequence { n k } of positive integers, such that n 1 < n 2 < n 3 < ··· . Then the sequence { p n i } is called a subsequence of { p n } . If { p n i } converges, its limit is called a subsequential limit of { p n } . 3.6 Theorem (a) If { p n } is a sequence in a compact metrix space X , then some subsequence of { p n } converges to a point of X . (b) Every bounded sequence in R k contains a convergent subsequence. 3.7 Theorem The subsequential limits of a sequence { p n } in a metric space X form a closed subset of X . 1 Cauchy Sequences 3.8 Definition A sequence { p n } in a metric space X is said to be a Cauchy sequence if for every ǫ > there is an integer N such that d ( p n , p m ) < ǫ if n ≥ N and m ≥ N ....

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