Advanced Calculus Homework 3 Page 1 of 2

Deadline : 2023/10/30, 17:00.

1. Let S1 and S2 be two nonempty subsets in a metric space with S1 ∩ S2 = S2 ∩ S1 = ∅. If

A ⊆ S1 ∪ S2 is a connected set, then either A ⊆ S1 or A ⊆ S2 .

2. If A1 and A2 are two nonempty and connected sets with A1 ∩ A2 ̸= ∅. Prove or disprove
that

1. A1 ∩ A2 is connected.
2. A1 ∪ A2 is connected.

3. Let {Ak }∞k=1 be a family of connected subsets of M , and suppose that A is a connected
subset of M such that Ak ∩ A ̸= ∅ for all k ∈ N. Show that the union (∪k∈N Ak ) ∪ A is also
connected.
Pn
4. Let {ak }∞
k=1 be a sequence and define sn = n
1
k=1 ak . Prove or disprove that

1. If ak converge, then sn converge.

2. If sn converge, then ak converge.
(2n−1)a1 +(2n−3)a2 +···+3an−1 +an
3. Let tn = n2
. Assume ak converge to a. Does tn also converge
to a ?

5. If ak > 0 for all k ∈ N, prove that

ak+1 √ √ ak+1
lim inf ≤ lim inf k ak ≤ lim sup k ak ≤ lim sup
k→∞ ak k→∞ k→∞ k→∞ ak

√ ak+1
Moreover, find a {ak }∞
k=1 such that lim sup
k
ak < lim sup
k→∞ k→∞ ak
√
6. If s1 = 2, and
p √
sn+1 = 2 + sn (n = 1, 2, 3, · · · ),

prove that sn converges, and that sn < 2 for n = 1, 2, 3, · · · .

P P
7. Suppose an > 0 and sn = nk=1 ak . If sn diverge. Prove or disprove that tn = nk=1 ak
1+ak
diverges. What can be said about
P
1. Sn = nk=1 1+ka
ak
k
.
P
2. Tn = nk=1 1+kak2 ak .
P P
3. If sn = nk=1 ak converge. Does Jn = nk=1 kak converge.

8. Assume A ⊂ R is compact and let a ∈ A. Suppose {an } is a sequence in A such that every
convergent sub-sequence of {an } converges to a.

1. Does the sequence {an } also converge to a ?

2. Without the assumption of A is compact. Does the sequence {an } converge to a ?
 9. Suppose that ak ̸= 0 for large k and that

ln(1/|ak |)
p = lim
k→∞ ln(k)
P∞
existsP
as an extended real number. If p > 1, then k=1 ak converges absolutely. If p < 1,
then ∞ k=1 ak diverges.

10. Suppose that f : R → (0, ∞) is differentiable, that f (x) → 0 as x → ∞, and that

xf ′ (x)
α ≡ lim
x→∞ f (x)

P∞
exists. If α < −1, prove that k=1 f (k) converges.

11. Suppose that {an } is a sequence of nonzero real numbers and that

 
ak+1
p = lim k 1 −
k→∞ ak

P∞
exists as an extended real number. Prove that k=1 ak converges absolutely when p > 1.

Extra question
(If you finish there problems and want to obtain extra points, please email
symmetrickelly@gmail.com)

12. Please read, state and prove following theorem from William R Wade’s ”An Introduction
to Analysis” P.209 ∼ P.211

1. Abel’s Formula
2. Dirichlet’s Test
3. Leibniz’s criterion (Alternating series test)

13. Use Abel’s Formula directly prove Leibniz’s criterion (Consider S2n+1 and S2n and show
that they are monotone).
P∞ sin(k) P∞ cos(k)
14. Show that k=1 k
and k=1 k

15. Understand what is the sequence of function and what is the definition of the sequence of
function point-wise converge and uniformly converge.

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