Sequences
SUGGESTED REFERENCE MATERIAL:
As you work through the problems listed below, you should reference your lecture notes and
the relevant chapters in a textbook/online resource.
EXPECTED SKILLS:
• Find the general term of a sequence.
• Determine whether a sequence converges, and if so, what it converges to. This may
require techniques such as L’Hopital’s Rule and The Squeeze Theorem.
PRACTICE PROBLEMS:
For problems 1 – 8, rewrite the sequence by placing the general term inside
braces.
1.
2.
1 1 1 1
, , ,
, ...
4 16 64 256
+∞
1
. Note that any integer can be used as the lower index for a sequence. So
4n n=1
+∞
+∞
1
1
you could describe this sequence with
or even
, although
4n+1 n=0
4n+10 n=−9
admittedly it is weird to use −9 as a starting index unless there is some compelling
reason to do so. It is recommended to use either 0 or 1 as a starting index in most
cases.
1
1 1
1
,− , ,−
, ...
4 16 64 256
+∞
n+1 1
(−1)
4n n=1
3. 0, 1, 23 , 34 , 45 , ...
n+1 +∞
n
n=0
4. 3, 2, 1, 0, −1, −2, −3, −4, −5, ...
{4 − n}+∞
n=1
1
1
1
5. 1, , e2 , 3 , e4 , 5 , ...
e
e
e
(−1)n n +∞
e
n=0
1
6.
7.
1
1
1
1
,
,
,
, ...
1·2 1·2·3·4 1·2·3·4·5·6 1·2·3·4·5·6·7·8
+∞
1
(2n)! n=1
3
5
7
9
,
,
,
, ...
1·2 1·2·3·4 1·2·3·4·5·6 1·2·3·4·5·6·7·8
+∞
2n + 1
(2n)! n=1
8. 0, 1, 0, −1, 0, 1, 0, −1, 0, 1... [Hint: Think about a trigonometric function.]
n π o+∞
n
π o+∞
n π o+∞
sin
n
n
n
or − cos
or cos
. There are other possibilities as well.
2
2
2
n=0
n=1
n=−1
9. For each of the sequences in problems 1 – 8, determine if the sequence converges, and
if so, what it converges to. If it diverges, determine if the general term approaches
+∞, −∞, or neither.
The sequence in problem:
#1 converges to 0.
#2 converges to 0.
#3 diverges to +∞.
#4 diverges to −∞.
#5 diverges. The even-numbered terms converge to 0, but the odd-numbered
terms diverge to +∞
#6 converges to 0.
#7 converges to 0.
#8 diverges. The odd-numbered terms converge to 0 (in fact, they are all equal
to 0), but the even-numbered terms diverge since they oscillate between 1 and
−1.
Detailed Solution: Here
For problems 10 – 35, determine if the sequence converges, and if so, what it
converges to. If it diverges, determine if the general term approaches +∞, −∞,
or neither.
10. {5}+∞
n=1
Converges to 5.
2
11. {5n}+∞
n=0
Diverges to +∞.
12.
5 − 5n3
+∞
n=1
Diverges to −∞.
13.
4n − 3n5
2n5 + 4n3 + n2 + 5
+∞
n=1
3
Converges to − . See Limits at Infinity Review problem #2.
2
+∞
3
2
n n +n +n+1
14. (−1)
n3 + 1
n=1
The odd-numbered terms converge to −1, but the even-numbered terms converge to
1, so the sequence diverges.
15.
n4 − 3n3 − 2n
4n2 + 19
+∞
n=0
Diverges to +∞.
16.
1 − 10n2
n2 − 4n3
+∞
n=1
Converges to 0.
2 +∞
n+1 1 − 10n
17. (−1)
n2 − 4n3 n=1
Converges to 0.
(√
18.
4 + 3n2
2 + 7n
)+∞
n=1
√
3
Converges to
. See Limits at Infinity Review problem #3.
7
+∞
19. e1/n n=1
Converges to 1.
3
20.
e−n
n−2
+∞
n=1
Converges to 0.
21.
+∞
en − e−n
en + e−n
n=1
Converges to 1. See Limits at Infinity Review problem #5.; Detailed Solution: Here
22.
√
e
n
+∞
n
n=1
Diverges to +∞.
23.
+∞
en sin(e−n ) n=1
Converges to 1.
24.
en π −n
+∞
n=1
Converges to 0.
+∞
1
25. ln
n
n=1
Diverges to −∞.
26.
ln (6n)
ln (2n)
+∞
n=1
Converges to 1.; Detailed Solution: Here
27. {ln (n + 2) − ln (3n + 5)}+∞
n=0
1
Converges to ln
.
3
28.
n√
n2
o+∞
+ 8n − 5 − n
n=1
Converges to 4. See Limits at Infinity Review problem #4.; Detailed Solution: Here
29.
n√
n2
−n+n
o+∞
n=0
Diverges to +∞.
4
30.
n√
o+∞
n2 − n − n
n=1
31.
1
Converges to − .
2
n cos n o+∞
n
n=1
Converges to 0. See Limits at Infinity Review problem #7.
32. arccos
n2
3n − n2
+∞
n=1
Converges to π.
+∞
1
33. arctan
− arctan (n)
n
n=1
π
Converges to − . See Limits at Infinity Review problem #8.
2
n +∞
1
34.
1+
n
n=1
Converges to e. See Limits at Infinity Review problem #9.
n
o+∞
n 1/n
35. (1 + 3 )
n=1
Converges to 3. See Limits at Infinity Review problem #10.; Detailed Solution: Here
( )+∞
2/n
4
36.
n
n=1
Converges to 1.
37. Consider the sequence
√
r
q
q
√
√
30, 30 + 30, 30 + 30 + 30, ...
(a) Define the sequence recursively.
√
√
a1 = 30, an+1 = 30 + an for integers n ≥ 1.
(b) Assuming the sequence converges to some limit L, find L.
L = 6.
38. Consider the sequence {an }+∞
n=1 that has the following recursive definition:
an+1 = 10 − an for integers n ≥ 1.
5
(a) Assuming the sequence converges to some limit L, find L.
L = 5.
(b) How must a1 be defined to ensure that the sequence converges? Justify your
answer.
If a1 = 5, the sequence is 5, 5, 5, 5, ..., which clearly converges to 5. If a1 6= 5,
say a1 = K(K 6= 5), then the sequence oscillates bewteen K and 10 − K, e.g. if
a1 = 3 we have 3, 7, 3, 7, 3, 7, ... Such a sequence diverges.
39. The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, ... begins with two 1’s and thereafter each
term in the sequence is the the sum of previous two terms.
(a) Define the Fibonacci sequence recursively.
a1 = 1, a2 = 1, an+2 = an + an+1 for integers n ≥ 1.
(b) Clearly the Fibonacci sequence diverges to +∞, but consider the ratio of succesan+1
sive terms
for n ≥ 1, i.e
an
1 2 3 5 8
, , , , , ...
1 1 2 3 5
Assuming this “ratio sequence” converges to some limit L, find L.
√
1+ 5
L=
. This number is known as the Golden Ratio.
2
Fibonacci Fun Fact: The Fibonacci Sequence can be described without recursion
as
 √ n √ n +∞
 1+2 5 − 1−2 5 
√
,


5
n=1
and with some clever factoring it can be shown from
√ this non-recursive definition
1+ 5
that the limit of the “ratio sequence” is L =
.
2
Detailed Solution: Here
6