Math 112 Worksheet 3: Infinite Sequences
1. Write out the first 5 terms of the following sequences, starting at a1 .
(n + 3)!
n(n + 2)!
an−1 + n
a1 = 1, an =
n−1
a1 = 3, a2 = 1, an = |an−1 − an−2 |
n
an = (−1)n+1
n!
(n − 1)!(n + 5)!
an =
n!(n + 3)!
(a) an =
(b)
(c)
(d)
(e)
2. Find a formula given the first few terms of the given sequences. Just for fun, try to write a
different formula, either by simply shifting n or via algebraic manipulations.
1
(a) 3, 1, 13 , 19 , 27
,...
(see back side)
(b) 1, −1, 1, −1, 1, . . .
4 16
32
(c) −1, 52 , − 25
, 125 , − 625
1
(d) − 13 , 15 , − 71 , 91 , − 11
,...
3. Determine whether the following sequences converge or diverge. If it converges, say what it
converges to.
(a) an = n 2−n
n!
(b) an = 2
n πn + 1
(c) an = cos
n
(d) an = arctan(en )
(e) an =
cos2 n
n+1
Challenge problems: Find a formula for the sequences given below, assuming the pattern continues.
1. 1, 3, 6, 10, 15, 21, . . .
2. −1, −1, 1, 1, −1, −1, . . .
3. 1, 5, 12, 22, 35, 51, . . .
Definition & Remarks A geometric sequence is a sequence of the form a, ar, ar2 , ar3 . . ., where
a, r are fixed real numbers. For examples, 3, 32 , 34 , . . .. Here a = 3, r = 12 . We use the letter r
because we sometimes refer to this number as the fixed ratio of the geometric sequence (notice that
such a sequence will only converge if |r| < 1, so we usually think of it as a fraction). Given two
terms of a geometric sequence, it is easy to recover the formula for an = arn since a0 = a and
an
arn
= n−1 = r.
an−1
ar
Solutions
1. 4, 5/2, 2, 7/4, 8/5, . . .
2. 1, 3, 3, 7/3, 22/12, . . .
3. 3, 1, 2, 1, 1, . . .
4. 1, −1, 1/2, −1/6, 1/24, . . .
5. 30, 21, 56/3, 18, 18, . . .
( )∞
∞
1
1 n−1
6.
or 3
3n−2 n=1
3
n=1
or
1
3n−1
∞
n=0
n+1 }∞ or {(−1)n−1 }∞ or {−(−1)n }∞
7. {(−1)n }∞
n=0 or {(−1)
n=1
n=1
n=1
)
(
∞
n ∞
n−1
n ∞
2
2
2
8. (−1)n+1
or (−1)n
or − −
5
5
5
n=0
n=0
n=1
9.
∞
(−1)n−1
(−1)n ∞
or
2n + 1 n=1
2n − 1 n=2
10. Converges to 0
11. Diverges
12. Converges to −1
13. Converges to π/2
14. Converges to 0
Challenge problems:
n
o∞
1. n(n+1)
(These are the triangular numbers)
2
n=1
n
o
n(n+1) ∞
2. (−1) 2
n=1
3.
n
3n2 −n
2
o∞
n=1
(These are the pentagonal numbers)