MATH231 Lecture Notes 3

MATH231 Lecture Notes 3 - 1 Sequences and Series Infinite...

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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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MATH231 Lecture Notes 3 - 1 Sequences and Series Infinite...

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