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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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 Spring '08
 Bronski
 Math, Calculus, Infinite Series, Sequences And Series, lim, 2k, partial sums

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