Math 152-copyright Joe Kahlig, 13a
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Section 10.1: Sequences
A sequence is a list of numbers written in a definite order.
a1 , a2 , a3 , .... or {an }∞
n=1
Example: Find a general formula for these sequences.
A)
B)
5 6 7 8
, , , , ...
9 16 25 36
6 9
15
1, , , 2, , ...
4 5
7
C) {1, 1, 2, 3, 5, 8, 13, 21, ...}
Definition: A sequence {an } is said to have the limit L, written lim an = L or an → L as n → ∞,
n→∞
if we can make the terms an as close to L as we like by taking n sufficiently large. If lim an exists,
n→∞
we say that the sequence converges (or is convergent). Otherwise, we say the sequence diverges (or is
divergent).
Definition: If {an } is a sequence, then lim an = L means that for every ǫ > 0 there is a correspondn→∞
ing integer N such that |an − L| < ǫ whenever n > N .
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Math 152-copyright Joe Kahlig, 13a
Example: Do these sequences converge or diverge.
A) {(−1)n }∞
n=1
B) {cos(2nπ)}
C)
3n
n+2
∞
n=5
THEOREM If lim f (x) = L and f (n) = an when n is an integer, then lim an = L.
x→∞
n→∞
Example: Does the sequences converge or diverge? If it converges, give the value.
A)
(
n2
ln(3 + en )
)
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Math 152-copyright Joe Kahlig, 13a
B)
(
n2
3n
+ 2
n+2 n +1
)
Limit Laws for Convergent Sequences: If {an } and {bn } are convergent sequences and c is a
constant, then
lim an bn = lim an ∗ lim bn
lim (an + bn ) = lim an + lim bn
n→∞
lim (an − bn ) = lim an − lim bn
an
n→∞ bn
n→∞
n→∞
n→∞
n→∞
n→∞
n→∞
lim
n→∞
n→∞
lim an
=
lim bn ,
n→∞
n→∞
if lim bn 6= 0
n→∞
lim can = c lim an
n→∞
n→∞
Squeeze Theorem for Sequences: If an ≤ bn ≤ cn for n ≥ no and lim an = lim cn = L, then
n→∞
n→∞
lim bn = L
n→∞
Example: Does the sequences converge or diverge? If it converges, give the value.
A)
(
(−1)n n2
n2 + 1
)
Math 152-copyright Joe Kahlig, 13a
B)
(−1)n 3n
n2 + 5
C)
n!
nn
Example: Find the values of r so that {r n } converges.
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Math 152-copyright Joe Kahlig, 13a
Definition A sequence {an } is called increasing if an < an+1 for all n ≥ 1, that is a1 < a2 < a3 < ....
It is called decreasing if an > an+1 for all n ≥ 1. A sequence is monotonoic fi it is either increasing
or decreasing.
Example: Show that the sequence an =
n2
n
is a decreasing sequence.
+4
Definition A sequence {an } is bounded above if there is a number M such that an ≤ M for all
n ≥ 1. It is bounded below if there is a number m such that m ≤ an 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.
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Math 152-copyright Joe Kahlig, 13a
Example: A sequence is given by a1 =
√
5, an+1 =
√
5 + an
(A) By induction or other methods, show that an is increasing and bounded above by 4
(B) Find lim an
n→∞
Math 152-copyright Joe Kahlig, 13a
Example: A sequence is given by a1 = 1, an+1 = 3an − 1
(A) Show that this is an increasing sequence.
(B) Does this sequence converge?
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