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The solutions to problem 1-3 of the week #8 practice quiz for math 412. The problems deal with determining the values of z for which the given power series converge, finding the radius of convergence of specific power series, and proving the relationship between the radii of convergence of related power series.
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Problem 1. (6 points) Determine all values of z such that
โ^ โ
n=
z 1 + z
)n
converges, and for points of convergence, determine the sum.
Problem 2. (4 points each) Determine the radius of convergence of each power series below:
(a): โโ
n=
n! 1000 n
zn
(b): โโ
n=
n^3 (n + 1)n (3n)n
(z โ i + 3)n.
Problem 3. (6 points) Assume that the radius of convergence of the power series
โ^ โ
n=
cnzn
is ฯ, where 0 < ฯ < โ. Prove that the radius of convergence of the power series
โ^ โ
n=
c^2 nzn
is ฯ^2.
