2. First do Problem #24 on p. 163 (as usual, using Problem 5 below to justify replacing roots by ratios in the definition of the radius of convergence). Then note that since the power series in question has radius of convergence 1, we can replace the real variable x with a complex variable z and then use the power series as the definition of(1+z)^a for z = x + iy and
|z| = sqrt(x2+y2) < 1. But with a = 1/n for n a positive integer, there are n complex numbers w for which w = 1 + z. Which of these is described by the power series?
[You may want to dwell on the case n = 2 where the two square roots differ only in sign and use this case to guess an answer to the above question for bigger n's].
Hint: There are lots of ways to address this question. One way uses connectivity ideas.
24. Let a R, a 0, 1, 2 Show that the "binomial series" ......
has radius of convergence 1. Let f (x) be the sum of this series on its interval of convergence. Show that (1 + x)f'(x)= af(x), and hence thatf(z) (1 + x)^a for x < 1.
This problem is from Introduction to Analysis (Maxwell Rosenlicht)
See attached file for full problem description.
Complex Power Series and Radius of Convergence are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.