1120_10 - Week 10 Outline Denition of sequence Limit of...

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Week 10 Outline Definition of sequence Limit of sequence Six limits to remember Monotonic sequences Bounded sequences Bounded, monotonic sequence is convergent Definition of series –Convergence and divergence of series –Geometric series –Telescoping series Divergence test A sequence can be considered as a list of numbers written in a definite order: a 1 , a 2 , a 3 , . . . , a n , . . . , where n is the index of a n indicating where a n occurs in the list. Remark A sequence can be defined as a function n 7→ a n with domain { 1 , 2 , . . . } . Definition An infinite sequence of numbers is a function whose domain is the set of positive integers. Examples Some sequences can be defined by giving a formula for the n th term. 1. { n n +1 } n =1 2. { ( - 1) n ( n +1) 3 n } n =1 3. { n - 3 } n =3 = { n } n =0 4. { cos 6 } n =0 5. { p n } , where p n is the population of the world as of January 1st in the year n . 6. { f n } , where f 1 = 1, f 2 = 1, f n = f n - 1 + f n - 2 , for n 3. This sequence answers the following question. Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one new-born pair, how many pairs of rabbits will we have in the n th month? 1
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Definition A sequence { a n } has the limit L and we write lim n →∞ a n = L, or a n L, as n → ∞ if we can make a n as close to L as we like by taking n sufficiently large. If lim n →∞ a n exists, we say the sequence converges (or is convergent), otherwise we say the sequence diverges (or is divergent). If lim n →∞ a n = , we say that the sequence diverges to . If lim n →∞ a n = -∞ , we say that the sequence diverges to -∞ . If lim n →∞ a n simply does not exist (but is not or -∞ ), we say that the sequence diverges. For example, the sequence { 1 n } converges to 0, { n } diverges to , {- n } diverges to -∞ , and { ( - 1) n } diverges.
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