Calculus I Spring 2025 Homework 1
1. Prove that
n
p1
n
o1
is convergent by de…nition and …nd its limit.
n=1
2. For each following sequence fan g1
n=1 , prove it converges or diverges. If it converges,
…nd its limit.
6
(a) an = cos n(4n) . (Hint: Use Squeeze Theorem)
6
n
(b) an = n4 +1989
.
n
(c) an = 2n! . (Hint: Use Squeeze Theorem)
p
p
n. (Hint: Use Squeeze Theorem and the result of 1)
(d) Fix k 2 N. an = n + k
m
m
X p
X
(e) Fix m 2 N. an =
bk n + k, where
bk = 0. (Hint: Use the result of 2(d))
k=1
k=1
3. Let fan g1
n=1 be a sequence.
(a) Prove that lim an = L , lim (an
n!1
L) = 0.
n!1
(b) Prove that lim an = 0 , lim jan j = 0.
n!1
n!1
1
4. Let fan g1
n=1 and fbn gn=1
o convergent sequences with limits A and 0 respectively. If
n be
bn 6= 0, 8n 2 N and if
an
bn
1
is convergent, prove that A = 0.
n=1
5. Suppose that lim an = L, for some L 2 R.
n!1
the de…nition of limit)
Prove that lim n1
n!1
n
X
ak = L. (Hint: Use
k=1
6. Let f (x) = 6 14jxj .
(a) Find the domain and range of f .
(b) Sketch the graph of y = f (x). Indicate each steps that how you get the graph of
y = f (x) from the graph of y = x1 .
7. Do the following exercises in the textbook:
1.1: 40, 42, 43, 44, 47, 53.
1.3: 6, 7, 31.
11.1: 29, 31, 34, 41, 45 (Hint: Prove by contradiction and use the result of exercise
11.1.76(a)), 61 (Hint: Use the result of 2(c) and prove by contradiction), 62,
76(a)
1