MATH
1120_10

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

• Notes
• CommodoreFox15
• 7

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern