Math 101 – WORKSHEET 22
SEQUENCES
1. Skill 1: expression for sequences
(1) For each of the following sequences, write a formula for the general term
(a) {1, 2, 3, 4, 5, 6, · · · }
(b)
1 1 1 1 1
1, 4 , 9 , 16 , 25 , 36 , · · ·
(c) {3, 7, 11, 15, 19, · · · }
(d)
7
7
7
7
7
7
9 , 27 , 81 , 243 , 729 , 3187 , · · ·
(e)
1
1 3 1 5
3
7
1
9
5
2 , 2 , 8 , 4 , 32 , 32 , 128 , 32 , 512 , 512
···
=
1
2 3 4
5
6
7
8
9
10
2 , 4 , 8 , 16 , 32 , 64 , 128 , 256 , 512 , 1024 , · · ·
(f) {1, −1, 1, −1, 1, −1, 1, −1, 1, −1, 1, −1, · · · }
(g)
3 2 5 4 7 6 9 8 11
0, 8 , 27 , 64 , 125 , 216 , 343 , 512 , 729 , 1000 , · · ·
2. Skill 2: limits of sequences
(2) Determine
∞ if the sequences is convergent of divergent. If convergent, evaluate the limit.
(a) n1 n=1
(b)
n
n
n+1
o∞
n=1
∞
(c) {sin(n)}n=5
(d)
sin( n1 )
∞
n=1
Date: 2/3/2016, Worksheet by Lior Silberman. This instructional material is excluded from the terms of UBC Policy 81.
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(3) Further problems
(a) Does limn→∞
(b) limn→∞
n
2n
√ n
n+1000
exist?
=
∞
(c) (Math 103 final, 2014) Consider the sequence {an }n=1 = 1, 0, 21 , 0, 0, 31 , 0, 0, 0, 14 , 0, 0, 0, 0, 51 , · · · .
Decide whether limn→∞ an = 0.
3. Tool: Squeeze Theorem
(4) Determine if the sequences
is convergent
of divergent. If convergent, evaluate the limit.
∞
1
n
(a) (Final 2013) (−1) sin n n=1 .
(b) (Final 2011)
n
sin(n)
log(n)
o∞
(why do we have n ≥ 2 here?)
n=2
(c) (Math 105 Final 2012) an = 1 +
n! sin(n3 )
(n+1)! .
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