Into the largest and smallest accumulation points of

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

View Full Document Right Arrow Icon
into the largest and smallest accumulation points of { r a n } . If r < 0 then the multiplication transfers the largest accumulation point of { a n } into the smallest accumulation point of { r a n } , and the other way round. Warning. It may well happen that lim sup( a n + b n ) 6 = lim sup a n + lim sup b n and/or lim inf( a n + b n ) 6 = lim inf a n + lim inf b n . 1
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
SERIES Series is just an infinite sum of the form n =1 a n or a 1 + a 2 + a 3 + . . . , where a n R . Definition. We say that the series converges and write n =1 a n = c if the se- quence of finite partial sums s k = k n =1 a n converges to c . Equivalently ε > 0 k ε : k > k ε k X n =1 a n - c < ε. We say that a series is divergent if it is not convergent. Example. The series 1 - 1 + 1 - 1 + 1 . . . does not converge because its partial sums oscillate between -1 and 1. Example. An infinite decimal fraction is an example of a convergent series. In- deed, when we write a = 0 .a 1 a 2 a 3 . . . we actually mean that a = n =1 a n 10 - n . Remark. The series n =1 a n converges if and only if the series n = m a n con- verges. In other words the first few terms do not affect convergence, even though they change the value of the sum. Indeed, if s k are the partial sums of the original series then the partial sums σ k of the series n = m a n are equal to s k - a where a = m - 1 n =1 a n . From the definition of convergence for sequences, it follows that the sequence { s k } converges if and only if the sequence { σ k } converges. Example (geometric progression). If | b | < 1 then the series n =0 b n converges and n =0 b n = (1 - b ) - 1 . Indeed, by induction in m we obtain m X n =0 b n = (1 - b ) - 1 - b m +1 (1 - b ) - 1 . If | b | < 1 then b m +1 0 and, consequently, m X n =0 b n (1 - b ) - 1 as m → ∞ . Theorem. If the series n =1 a n converges then lim n →∞ a n = 0.
Image of page 2
Image of page 3
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