Math 152 – Spring 2016
Section 10.1
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Section 10.1 – Sequences
A sequence is a list of numbers written in a definite order:
a1 , a2 , a3 , . . . , an , . . .
Notation. The sequence {a1 , a2 , a3 , . . .} can also be written
{an },
{an }∞
n=1 ,
or
an = f (n).
Examples of Sequences. (with different notations shown)
1 ∞
an = n1 ,
1. 1, 21 , 13 , 14 , . . . , n1 , . . . ,
n n=1
n
o
n
n
(−1) (n+2)
4 −5
6
2. −3
,... ,
an = (−1) 5n(n+2) ,
5 , 25 , 125 , 625 , . . . ,
5n
√ √
√
0, 1, 2, 3, . . . , n − 3, . . . ,
n √
o
4. 1, 23 , 12 , 0, . . . , cos πn
6 ,... ,
3.
an =
√
n − 3, n ≥ 4,
an = cos
πn
6
n
(−1)n (n+2)
5n
√
∞
n − 3 n=4
πn
6
, n ≥ 0,
cos
o
∞
n=0
Example 1. Find a formula for the general term an of the sequence
5 −7 9 −11 13
,
, ,
,
,...
4 16 64 256 1024
Examples of Sequences without a simple defining equation:
1. Let an be the nth prime number. Then {an } is a well-defined sequence with first
terms
{2, 3, 5, 7, 11, 13, 17, 19, . . .}
2. The Fibonacci Sequence {fn } is defined recursively.
f1 = 1,
f2 = 1,
and
fn = fn−1 + fn−2 ,
for
n≥3
Then the sequence is
{1, 1, 2, 3, 5, 8, 13, 21, . . .}
There is a complicated formula for the Fibonacci Sequence:
φn − (1 − φ)n
√
f (n) =
5
where
√
1+ 5
φ=
.
2
Math 152 – Spring 2016
Section 10.1
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Graphs. A sequence can be graphed by plotting its terms on a number line or in a
2-dimensional plane.
Example 2. Example: an =
n
n+1
Real Definition. If {an } is a sequence, then
lim an = L
n→∞
means that for every > 0 there is a corresponding integer N such that
if
n > N,
|an − L| < .
then
Intuitive Definition. A sequence {an } has the limit L and we write
lim an = L
n→∞
or
an → L as n → ∞
if we can make the terms an as close to L as we like by taking n to be sufficiently large.
If limn→∞ exists, we say that the sequence converges (or is convergent). Otherwise,
we say the sequence diverges (or is divergent).
What is the difference between
lim an = L
n→∞
and
lim f (x) = L
x→∞
?
Theorem. If limx→∞ f (x) = L and f (n) = an , when n is an integer, then
lim an = L
n→∞
Example 3. Determine if {ne−n } converges or diverges.
Math 152 – Spring 2016
Section 10.1
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Infinite Limits. If an becomes arbitrarily large as n becomes large, then we use the
notation
lim an = ∞.
n→∞
But the sequence is still considered divergent.
Limit Rules: If {an } and {bn } are convergent sequences and c is a constant, then the
following are true.
1. lim (an + bn ) = lim an + lim bn
n→∞
n→∞
n→∞
2. lim can = c lim an
n→∞
n→∞
3. lim (an bn ) = lim an · lim bn
n→∞
n→∞
n→∞
limn→∞ an
an
=
,
bn
limn→∞ bn
h
ip
5. lim (an )p = lim an
4. lim
n→∞
n→∞
n→∞
if
if
lim bn 6= 0
n→∞
p>0
and
an ≥ 0
Squeeze Theorem for Sequences. If an ≤ bn ≤ cn for n ≥ no and
lim an = lim cn = L
n→∞
n→∞
then
lim bn = L
n→∞
Corollary. If limn→∞ |an | = 0, then limn→∞ an = 0
Example 4. Determine whether the sequences converge or diverge.
(a) an =
√
n
√
1+ n
(b) an =
ln n
n
Math 152 – Spring 2016
(c) an =
Section 10.1
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(−1)n n!
nn
Example 5. For what values of r is the sequence {rn } convergent?
Theorem. The sequence {rn } is convergent if −1 < r ≤ 1 and divergent for all other
values of r.


0 if −1 < r < 1
lim rn =
n→∞
 1 if
r=1
Math 152 – Spring 2016
Section 10.1
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Definition. A sequence {an } is called increasing if an < an+1 for all n ≥ 1, that is
a1 < a2 < a2 < · · · < an · · · .
It is called decreasing if an > an+1 for all n > 1, that is a1 > a2 > a3 > · · · > an > · · · .
A sequence is monotonic if it is either increasing or decreasing.
Example 6. Determine if the sequence is increasing, decreasing, or not monotonic.
(a) an = 3 +
(b) an =
(−1)n
n
n
n2 +1
Definition. A sequence {an } is bounded above if there is a number M such that
an ≤ M
for all
n ≥ 1.
A sequence {an } is bounded below if there is a number M such that
an ≥ M
for all
n ≥ 1.
If it is bounded above and below, then {an } is a bounded sequence.
Monotonic Sequence Theorem. Every bounded, monotonic sequence is convergent.
Example 7. Determine if the following sequences are convergent or divergent.
(a) an =
4
2n+7
Math 152 – Spring 2016
(b) a1 = 1 and an+1 =
Section 10.1
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1
6−3an
Principle of Mathematical Induction. Let Sn be a statement about the positive
integer n. If
1. S1 is true.
2. Si+1 is true whenever Si is true.
Then Sn is true for all positive integers n.
Application to Sequences: If the following are true
1. a1 < a2 and A ≤ a1 ≤ B
2. ai < ai+1 and A ≤ ai ≤ B whenever ai−1 < ai and A ≤ ai−1 ≤ B
Then an is a bounded, increasing sequence by mathematical induction. (Explanation:
Let Statement Si be that ai < ai+1 and A ≤ ai ≤ B.)
Similarly, if the each < is replace by >, then an is a bounded, decreasing sequence.
Solution for a1 = 1 and an+1 =
1
6−3an
above: