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Unformatted text preview: 1 Sequences and Series Infinite Series: A concept related to the idea of a sequence is an (infinite) series. A series is an infinite sum of the form ∞ X n =1 a n We say that a series converges if the limit lim N →∞ N X n =1 a n exists (This is the sequence of partial sums). Note that this is exactly analogous to improper integrals, where we say that the integral converges if the limit lim R →∞ Z R 1 f ( x ) dx converges. Examples ∑ ∞ n =1 ar n (1) ∑ ∞ n =1 1 n (2) ∑ ∞ n =1 1 n ( n +1) (3) Example ∞ X n =2 1 n 2 Converges. Proof: Note that n 2 > n ( n 1) 1 n 2 < 1 n ( n 1) S N = N X n =2 1 n 2 < N X n =2 1 n 1 1 n = 1 1 N < 1 . S N is increasing and bounded, so the series converges. Theorem: If lim n →∞ a n 6 = 0 then ∞ X n =1 a n 1 does not converge. Proof: Define the partial sums S N = N X n =1 a n Note that lim N →∞ S N = L As well as lim N →∞ S N 1 = L Thus lim N →∞ S N S N 1 = 0 NOTE: lim n →∞ a n = 0 does NOT guarantee that ∞...
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This note was uploaded on 04/07/2008 for the course MATH 231 taught by Professor Bronski during the Spring '08 term at University of Illinois at Urbana–Champaign.
 Spring '08
 Bronski
 Math, Calculus, Infinite Series, Sequences And Series

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