{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

chapter4 - Chapter 4 Sequences and Series 4.1 Innite...

Info icon This preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 4. Sequences and Series 4.1 Infinite sequences An infinite sequence (or sequence) of real numbers is an infinite succession of numbers which is usually given by some rule. We shall denote an infinite sequence by a 1 , a 2 , a 3 , · · · , a n , · · · , and we shall often write the sequence as { a n } ; and for each n , the number a n is called a term of the sequence.
Image of page 1

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

View Full Document Right Arrow Icon
2 MA1505 Chapter 4. Sequences and Series 4.1.1 Example (i) The sequence 0 , 1 , 2 , · · · , n - 1 , · · · is defined by the rule a n = n - 1. (ii) The sequence 1 , 1 2 , 1 3 , · · · , 1 n , · · · is defined by a n = 1 n . (iii) If a n = ( - 1) n +1 ( 1 n ), the sequence is 1 , - 1 2 , 1 3 , - 1 4 , 1 5 , · · · . (iv) If a n = n - 1 n , the sequence is 0 , 1 2 , 2 3 , 3 4 , 4 5 , · · · . 2
Image of page 2
3 MA1505 Chapter 4. Sequences and Series (v) If a n = ( - 1) n +1 , the sequence is 1 , - 1 , 1 , - 1 , · · · . (vi) If a n = 3, the sequence is 3 , 3 , 3 , · · · 4.1.2 Limits of sequences A number L is called the limit of a sequence { a n } , if for sufficiently large n , we can get a n as close as we want to a number L . We write lim n →∞ a n = L , or simply, a n L , Note that the limit of a sequence { a n } is unique. 4.1.3 Convergent and divergent Not all sequences have limits. 3
Image of page 3

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

View Full Document Right Arrow Icon
4 MA1505 Chapter 4. Sequences and Series If { a n } has a limit, we say the sequence is convergent and { a n } converges to L . If { a n } does not have a limit, we say { a n } is diver- gent. 4.1.4 Example (i) 0 , 1 , 2 , 3 , · · · is divergent. (ii) 1 , 1 2 , 1 3 , · · · , 1 n , · · · is convergent, its limit is 0. (iii) 0 , 1 2 , 2 3 , 3 4 , 4 5 , · · · converges to 1. (since lim n →∞ n - 1 n = 1. See example 4.1.6 (ii).) (iv) 1 , - 1 , 1 , - 1 , · · · is divergent. (v) 1 , - 1 2 , 1 3 , - 1 4 , 1 5 , · · · . converges to 0. (vi) If c is any real number, c, c, c, · · · clearly con- 4
Image of page 4
5 MA1505 Chapter 4. Sequences and Series verges to c . Such a sequence is called a constant sequence. 4.1.5 Some Rules on Limits Let lim n →∞ a n = A , and lim n →∞ b n = B , with A and B real numbers. (1) Sum rule: lim n →∞ ( a n + b n ) = A + B . (2) Difference rule: lim n →∞ ( a n - b n ) = A - B . (3) Product rule: lim n →∞ ( a n b n ) = AB . (4) Quotient rule: lim n →∞ a n b n = A B , if B 6 = 0. Using the above rules, we obtain: 4.1.6 Example (i) lim n →∞ ( - 1 n ) = ( - 1) lim n →∞ 1 n = - 1(0) = 0. 5
Image of page 5

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

View Full Document Right Arrow Icon
6 MA1505 Chapter 4. Sequences and Series (ii) lim n →∞ n - 1 n = lim n →∞ (1 - 1 n ) = lim n →∞ 1 - lim n →∞ 1 n = 1 - 0 = 1. (iii) lim n →∞ 5 n 2 = 5 lim n →∞ 1 n · lim n →∞ 1 n = 5 · 0 · 0 = 0. (iv) lim n →∞ 4 - 7 n 6 n 6 + 3 = lim n →∞ 4 n 6 - 7 1 + 3 n 6 = 0 - 7 1 + 0 = - 7. (Note that all the four rules in (4.1.5) are used in this example.) 4.1.7 Sequence and function Let { a n } be a sequence. Suppose there is a function f ( x ) such that a n = f ( n ). If lim x →∞ f ( x ) = L , then lim n →∞ a n = L. 6
Image of page 6
7 MA1505 Chapter 4. Sequences and Series 4.1.8 Example Consider the sequence { a n } where a n = n + 1 - n . Then a n = f ( n ) where f ( x ) is the function x + 1 - x . So lim n →∞ a n = lim x →∞ ( x + 1 - x ) = lim x →∞ 1 x + 1 + x = 0 .
Image of page 7

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

View Full Document Right Arrow Icon
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern