Mathematics 220
Homework Set 10
Not collect
1. 12.4
2. 12.6
3. 12.8
4. 12.48
5. 12.10 (for (a) you may try induction)
6. 12.7 (the textbook has a solution to this question, but you should try to solve it
before reading the solution)
7. For each of the following prove or give a counter-example
(a) If {an } converges to L then {|an |} converges to |L|.
Hint: you need to use the inequality |x| − |y| ≤ |x − y| for any
x, y ∈ R.
(b) If {|an |} is convergent then {an } is convergent.
(c) an → 0 if and only if |an | → 0.
8. Find an example of each of the following
(a) A convergent sequence of rational numbers having an irrational limit.
Hint: You can use the fact that every real number has a decimal
expansion.
(b) A convergent sequence of irrational numbers having a rational limit.
9. Let {an } be a sequence of positive numbers. Prove that if {an } diverges to infinity
then {1/an } converges to 0.
10. Let {an }, {bn }, {cn } be sequences such that an ≤ bn ≤ cn for all n ∈ N. Prove that
if an → b and cn → b then bn → b.
11. Mark True or False. Justify each answer.
(a)
(b)
(c)
(d)
(e)
If
If
If
If
If
{an }
{an }
{an }
{an }
{an }
converges to a and an > 0 for all n ∈ N then a > 0.
and {bn } are both divergent sequences, then {an + bn } diverges.
and {bn } are both divergent sequences, then {an bn } diverges.
and {an + bn } are both convergent sequences, then {bn } converges.
and {an bn } are convergent, then {bn } is convergent.
12. Find the following limits
3n2 + 4n
(a) lim
n→∞ 7n2 − 5n
sin n
(b) lim
n→∞ 2n + 1
√
(c) lim ( n2 + 1 − n).
n→∞
13. Determine whether each series converges or diverges. Justify your answer.
∞
X
n5
(a)
2n
n=1
Mathematics 220
(b)
(c)
Homework Set 10
Not collect
∞
X
2n
n=1
∞
X
n=1
n!
1
(3n − 2)(3n + 1)
∞
X
√
√
(d)
( n + 1 − n)
n=1
14. Mark each statement True or False. Justify your answer.
P∞
(a)
n=1 an converges iff limn→∞ an = 0.
P
n
(b) The geometric series ∞
n=0 r converges iff r < 1.
P∞
(c)
an converges if the sequence {an } is bounded.
Pn=1
∞
(d)
n=1 an converges if the sequence {sn } where sn = a1 + ... + an is bounded.
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