lecture6 - 4 Power Series Sequences and series We say a...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
4. Power Series Sequences and series. We say a sequence s n C converges to s C if, given any ϵ > 0, there exists N N such that | s n s | < ϵ for all n N . The series k =0 z k (we don’t need to start at k = 0) converges if the sequence of partial sums s n = n k =0 z k converges. In this case, the limit of the sequence is called the sum of the series. A series which does not converge is said to be divergent. Absolute convergence. We say that n =0 z n is absolutely convergent if the real series n =0 | z n | is convergent. Exercise. n =0 z n is convergent if and only if n =0 Re z n and n =0 Im z n are both convergent. Suppose n =0 z n is absolutely convergent and write z n = x n + iy n . Then | x n | , | y n | ≤ | z n | so, by the Comparison Test, the real series n =0 x n , n =0 y n are absolutely convergent and hence convergent. Hence (by the exercise) n =0 z n is convergent.
Image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
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