Download Calc BC 10.2 sequences

Section 10.2
Sequences
In mathematics, a sequence is an ordered list of objects (or
events). Like a set, it contains members (also called elements
or terms), and the number of terms (possibly infinite) is
called the length of the sequence. Unlike a set, order
matters, and exactly the same elements can appear multiple
times at different positions in the sequence. A sequence is a
discrete function.
In this course we will only be interested in infinite sequences
with a rule for generating the terms (range values)
and using the set of whole numbers
or counting numbers as the domain.
Don’t let the new notation confuse you…
𝑓(𝑥) → 𝑎(𝑥) → 𝑎(𝑛) → 𝑎𝑛
1
1
𝑓 𝑥 = → 𝑎𝑛 =
𝑛
𝑥
𝒇 𝒙 𝐢𝐬 𝐜𝐨𝐧𝐭𝐢𝐧𝐮𝐨𝐮𝐬
𝒂𝒏 𝐢𝐬 𝐝𝐢𝐬𝐜𝐫𝐞𝐭𝐞
Other notation:
∞
𝑎𝑛 1
or simply just 𝑎𝑛
expanded form: 𝑎1 , 𝑎2 , 𝑎3 , … 𝑎𝑛 ,…
In our example:
1
𝑛
1 1 1
2 3 4
1
𝑛
= 1, , , , … , , …
What interests us most is what happens to the terms as 𝑛 → ∞
Since most discrete sequences have related continuous functions,
we will be using the same methods for
computing the limit as 𝑛 → ∞
as were used to evaluate limits of continuous functions in chapter 2.
(including L’Hopital’s Rule)
Given an infinite sequence 𝑎𝑛 , if lim 𝑎𝑛 = 𝐿 (a real number)
𝑛→∞
then the sequence is said to converge (to 𝐿).
If lim 𝑎𝑛 = ±∞ or does not exist,
𝑛→∞
we say the sequence diverges.
There are two main types of divergence:
1) Divergence to ∞ or − ∞ (ex: 1, 2, 3, 4, 5, …)
2) Divergence by oscillation (ex: 1, 2, 1 , 2, 1 , 2, …)
Some oscillating sequences
never settle down to
approach a single number.
The car converged to the tree.
The rocket is diverging to
… or so says NASA.
∞
Ex: #4 (read the directions)
Step 1: line up the most
reasonable domain #’s.
The numerators are all 1’s.
The denominators are
all powers of “2”.
We need the “universal
alternator” (−1)𝑛
If we started with n=1…
1,
0
1 1
1
− , ,− ,…
2 4
8
1 2
3 …
1
?
1
2𝑛
Final answer: 𝑎𝑛 =
(−1)𝑛+1
𝑎𝑛 =
2𝑛−1
(−1)𝑛
2𝑛
Ex: #20 (read the directions)
lim
𝑛→∞
2 + 𝑐𝑜𝑠𝑛
=?
𝑛
lim
𝑛→∞
𝑎𝑛 =
2 + 𝑐𝑜𝑠𝑛
𝑛
Since −1 ≤ 𝑐𝑜𝑠𝑛 ≤ 1,
The numerator of each
term is a constant
between 1 & 3.
1
2 + 𝑐𝑜𝑠𝑛
3
≤ lim
≤ lim
𝑛→∞
𝑛→∞ 𝑛
𝑛
𝑛
The limits on each end are = 0.
Final answer: this sequence converges to zero
by the Squeeze Theorem.
Ex: #30
𝑎𝑛 =
𝑛3
𝑒 𝑛/10
∞
lim
𝑛3
𝑛→∞ 𝑒 𝑛/10
Use L’Hopital’s Rule
∞
2
3𝑛
= ?lim
L’Hopital again…
𝑛→∞ .1𝑒 𝑛/10
∞
6𝑛
= lim
𝑛→∞ .01𝑒 𝑛/10
∞
One more time!
6
= lim
=0
𝑛/10
𝑛→∞ .001𝑒
This sequence converges to zero.
Ex: #38
2
𝑎𝑛 = 1 − 2
𝑛
𝑛
This is the indeterminate form 1∞ .
More precisely 1−
∞.
0 ≤ 𝑎𝑛𝑠𝑤𝑒𝑟 ≤ 1
𝑛
2
𝑙𝑛 1 − 2
𝑛
= lim
1
𝑛→∞
𝑛
0
0
2
2
lim 𝑙𝑛 1 − 2
= lim 𝑛 𝑙𝑛 1 − 2
𝑛→∞
𝑛→∞
𝑛
𝑛
1
−3
∙
4𝑛
2
1
4
1− 2
= lim
−
𝑛
2
𝑛→∞
𝑛
= lim
−2
1
−
𝑛→∞
−𝑛
𝑛2
−4
= lim
2
𝑛→∞
𝑛−
𝑛
−4𝑛
−4
= lim 2
= lim
=0
𝑛→∞ 𝑛 − 2
𝑛→∞ 2𝑛
Don’t forget to
anti-
This sequence converges to 𝑒 0 = 1
2
2𝑛 + 3
Ex: #44 (read the directions) 𝑎𝑛 =
→
5
5𝑛 − 17
Sequence mode is way too slow at
creating tables…so let’s switch back to
function mode!
Ex: #50
𝑎𝑛 =
3𝑠𝑖𝑛−1
3𝑛 − 1
→ 3𝑠𝑖𝑛−1
4𝑛 + 1
3
4
𝜋
=3
=𝜋
3
hmmm…looks like 𝜋 …
This sequence converges to 𝜋.
Are you thinking ≈ 3 ?