Question

Show that the series ∑ k=1+[infinity]​ 1−z k z k converges uniformly on D(0;r)

​ for every r∈(0,1), but diverges on C\ D(0;1)
​

Answer 1

The series converges uniformly on the disk D(0;r) for 0

Step-by-step explanation:

The question asks to demonstrate whether the series ∑ from k=1 to infinity of (1 - z^k)/z^k converges uniformly within the disk D(0;r) for any r in the interval (0,1), but diverges outside the closed disk D(0;1). To show uniform convergence within the disk D(0;r), one can use the comparison test and the Weierstrass M-test, noting that for any |z| < r < 1, the terms of the series are dominated by a geometric series that converges. Thus, since the geometric series converges uniformly, so does our original series. In contrast, for any point z on the complex plane outside the closed unit disk (|z| ≥ 1), the series diverges because the terms do not tend to zero as k goes to infinity.