PureMath.pdf

# We were concerned only with a real variable x we

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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.

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[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.]
[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

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