We were concerned only with a real variable x we

Info icon This preview shows pages 444–447. Sign up to view the full content.

View Full Document Right Arrow Icon
we were concerned only with a real variable x . We shall now consider a few general properties of power series in z , where z is a complex variable. A. A power series a n z n may be convergent for all values of z , for a certain region of values, or for no values except z = 0 . It is sufficient to give an example of each possibility.
Image of page 444

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

View Full Document Right Arrow Icon
[VIII : 193] THE CONVERGENCE OF INFINITE SERIES, ETC. 429 1. The series z n n ! is convergent for all values of z . For if u n = z n n ! then | u n +1 | / | u n | = | z | / ( n + 1) 0 as n → ∞ , whatever value z may have. Hence, by d’Alembert’s Test, | u n | is convergent for all values of z , and the original series is absolutely convergent for all values of z . We shall see later on that a power series, when convergent, is generally absolutely convergent. 2. The series n ! z n is not convergent for any value of z except z = 0 . For if u n = n ! z n then | u n +1 | / | u n | = ( n + 1) | z | , which tends to with n , unless z = 0. Hence (cf. Exs. xxvii . 1, 2, 5) the modulus of the n th term tends to with n ; and so the series cannot converge, except when z = 0. It is obvious that any power series converges when z = 0. 3. The series z n is always convergent when | z | < 1 , and never convergent when | z | = 1 . This was proved in § 88 . Thus we have an actual example of each of the three possibilities. 192. B. If a power series a n z n is convergent for a particular value of z , say z 1 = r 1 (cos θ 1 + i sin θ 1 ) , then it is absolutely convergent for all values of z such that | z | < r 1 . For lim a n z n 1 = 0, since a n z n 1 is convergent, and therefore we can certainly find a constant K such that | a n z n 1 | < K for all values of n . But, if | z | = r < r 1 , we have | a n z n | = | a n z n 1 | r r 1 n < K r r 1 n , and the result follows at once by comparison with the convergent geomet- rical series ( r/r 1 ) n . In other words, if the series converges at P then it converges absolutely at all points nearer to the origin than P . Example. Show that the result is true even if the series oscillates finitely when z = z 1 . [If s n = a 0 + a 1 z 1 + · · · + a n z n 1 then we can find K so that | s n | < K for all values of n . But | a n z n 1 | = | s n - s n - 1 | 5 | s n - 1 | + | s n | < 2 K , and the argument can be completed as before.]
Image of page 445
[VIII : 193] THE CONVERGENCE OF INFINITE SERIES, ETC. 430 193. The region of convergence of a power series. The circle of convergence. Let z = r be any point on the positive real axis. If the power series converges when z = r then it converges absolutely at all points inside the circle | z | = r . In particular it converges for all real values of z less than r . Now let us divide the points r of the positive real axis into two classes, the class at which the series converges and the class at which it does not. The first class must contain at least the one point z = 0. The second class, on the other hand, need not exist, as the series may converge for all values of z . Suppose however that it does exist, and that the first class of points does include points besides z = 0. Then it is clear that every point of the first class lies to the left of every point of the second class. Hence there is a point, say the point z = R
Image of page 446

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

View Full Document Right Arrow Icon
Image of page 447
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