MATH 131-CALCULUS I: HOMEWORK 4
Due: Wednesday, March 11
(1) Suppose there exists N0 such that sn ≤ tn for all n > N0 .
(a) Prove that if limn→∞ sn = +∞ then limn→∞ tn = +∞.
(b) Prove that if limn→∞ tn = −∞ then limn→∞ sn = −∞.
(c) Prove that if limn→∞ sn and limn→∞ tn both exist then limn→∞ sn ≤ limn→∞ tn .
(2) (a) Show that if limn→∞ sn = +∞ and k > 0, then limn→∞ ksn = +∞.
(b) Show that limn→∞ sn = +∞ if and only if limn→∞ −sn = −∞.
(c) Show that if limn→∞ sn = +∞ and k < 0 then limn→∞ ksn = −∞
(3) (a) Show that if limn→∞ sn = +∞ and inf{tn : n ∈ N} > −∞, then limn→∞ sn +
tn = +∞.
(b) Show that if limn→∞ sn = +∞ and limn→∞ tn > −∞, then limn→∞ sn + tn =
+∞.
(c) Show that if limn→∞ sn = +∞ and (tn ) is a bounded sequence, then limn→∞ sn +
tn = +∞.
(4) Assume all sn 6= 0 and that the limit L = limn→∞ | sn+1
| exists .
sn
(a) Show that if L < 1, then limn→∞ sn = 0.
(b) Show that if L > 1, then limn→∞ |sn | = ∞.
(5) Show

0



1
lim an =
∞
n→∞


 does not exist
if |a| < 1
if a = 1
if a > 1
if a ≤ −1.

 0
a
∞
lim
=
n→∞ np

does not exist
if |a| ≤ 1
if a > 1
if a ≤ −1.
(6) Show
n
(7) Show limn→∞
an
n!
= 0 for all a ∈ R.
1