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L14 Infinite Sequence - lim n!1 b n lim n!1 ca n = c lim n...

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Lecture 14: Sequences (I) Def. A sequence is a list of numbers written in a de±nite order: a 1 ;a 2 ;a 3 ;:::;a n ;::: Notation: f a 1 ;a 2 ;::: g = f a n g 1 n =1 = f a n g ex. f a n g = ± n n + 1 ² ex. f 1 ; ± 1 ; 1 ; ± 1 ;::: g 1
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(Note that elements of a sequence do not have to follow a pattern, for example: f a n g where a n = n th decimal place of e = 2 : 71 ::: . We are not interested in this kind of sequence here.) Limits of Sequences A sequence f a n g has the limit L if lim n !1 a n = L When the limit a n exists, we say that the sequence converges . If lim n !1 a n = 1 ; or if the limit does not exist, we say that the sequence diverges . 2
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Determine whether the sequence converges or di- verges. If it converges, ±nd the limit. ex. f ( ± 1) n g ex. ± 2 n 2 + 3 1 ± 3 n 2 ² ex. ± ( n ± 1)! ( n + 1)! ² 3
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Theorem: If f ( x ) is a function such that lim x ± > 1 f ( x ) = L and f ( n ) = a n ; where n is integer, then f ( x ) is called the related func- tion of the sequence f a n g and lim n ± > 1 a n = L . ex. ±² 1 + 1 n ³ n ´ , Convergent? ex. ± n sin ² 1 n ³´ , Convergent? 4
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NYTI: Do the sequences converge? ± ln n n ² ; f ne ± n g (0) Recall: Limit Laws: Let f a n g ; f b n g be convergent sequences and c a constant, then the result of the following operation is a convergent sequence and lim n !1 ( a n ² b n ) = lim n !1 a n ²
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Unformatted text preview: lim n !1 b n lim n !1 ca n = c lim n ± > 1 a n (lim c = c ) lim n !1 a n b n = lim n !1 a n lim n !1 b n lim n !1 a n b n = lim n !1 a n lim n !1 b n ; lim n !1 b n 6 = 0 lim n !1 ( a n ) p = ( lim n !1 a n ) p ;p > ;a n > 5 Squeeze Theorem: If a n ± b n ± c n (for n ² n ) and lim a n = lim c n = L , then lim b n = L . Determine if each sequence is convergent, if so, ±nd the limit: ex. ± cos 2 n 2 n ² . Convergent? ex. ± n ! n n ² . Convergent? 6 Absolute Convergence Thm: If lim j a n j = 0 ; then lim a n = 0 : ex. ± ( ± 1) n ± 1 n n 2 + 1 ² ex. ± sin n n 2 ² 7 Theorem: f r n g is convergent if ± 1 ² r ² 1 ; divergent otherwise. lim n !1 r n = 8 > < > : if j r j < 1 1 if r = 1 divergent if j r j > 1 ex. f a n g = ± 2 n 3 n +1 ² . Convergent? If so, ±nd the limit Tonight’s Homework: work out sequence prob-lems. 8...
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