Homework 4
Assignment: In Rudin read pages 52 to 55 (Cauchy Sequences), pages 57 to
72 (but you can skip the stuff on e on page 64, the stuff on “Summation by
Parts” on 70-71), and my handout on the Lim Inf and Lim Sup.
Do problems 6(a,b), 16, 20 in Chapter 3, and the problems below:
Additional Exercises
1. Find the radius of convergence of each of the power series
∑
(a)
2k xk
k
∑
k!xk
∑
√
(c)
xk / k!
∑
(d)
2k x3k
(b)
2. Suppose that lim sup ak and lim sup bk are both finite. Show that
lim sup(ak + bk ) ≤ lim sup ak + lim sup bk .
Given an example to show that strict inequality is possible.
∑ √x
∑
3. Prove that if k xk converges then k k k converges, if xk ≥ 0.
4. Let xk be a nonincreasing (xk ≥ xk+1 )∑sequence of positive numbers
∞
that converges to zero and suppose that k=1 xk converges. Prove that
limk kxk = 0. Show this result is false if the nonincreasing assumption is
dropped.
5. (Fun Extra Credit): Let S be a subset of R. Let C denote the operation
of taking the complement of S and K denote the operation of taking the
closure of S. For example, C([0, 1)) = R\[0, 1) = {x ∈ R; x < 0 or x ≥ 1}
and K([0, 1)) = [0, 1].
• Show that for any S we have C(C(S)) = S. That is, C 2 = I where
I is the “identity operator”.
• Show that for any S we have K(K(S)) = K(S), that is, K 2 = K.
• Starting with an arbitrary set S ⊆ R, what is the maximum number
of different sets that can be produced by applying C and K successively, in any order you like? Can you find a set S that actually
attains this maximum?
1