Unit 7 Notes:
Sequences & Series (with Limits)
Learning Target 1-LI: Calculate limits at finite numbers and infinity.
1-LI
Concept Notes: What is a limit?
Examine the limit as 𝑥 approaches 3 for 𝑓(𝑥) =
Graphically:
𝑥 2 −9
𝑥−3
.
Numerically (Table):
From the LEFT
From the RIGHT
Algebraically:
Concept: Limits to infinity.
Evaluate the following limits:
Graphically:
lim 𝑔(𝑥)
𝑥→−∞
and
lim 𝑔(𝑥)
𝑥→∞
Numerically (Table):
From the LEFT
From the RIGHT
Algebraically:
if 𝑔(𝑥) =
3𝑥 2 −27
𝑥 2 −4𝑥+4
Ex. 1: Evaluate the following given the graph of the function 𝐻(𝑥) below.
a. 𝐻(−4)
b.
e. 𝐻(1)
f.
i. 𝐻(6)
lim 𝐻(𝑥)
lim 𝐻(𝑥)
d. lim 𝐻(𝑥)
𝑥→−4 −
c.
lim 𝐻(𝑥)
g. lim+ 𝐻(𝑥)
h.
j. lim− 𝐻(𝑥)
k. lim+ 𝐻(𝑥)
l. lim 𝐻(𝑥)
𝑥→1−
𝑥→6
𝑥→−4 +
𝑥→1
𝑥→6
𝑥→−4
lim 𝐻(𝑥)
𝑥→1
𝑥→6
Ex. 2: Evaluate the following limits.
a.
𝑥2 − 5
lim
𝑥→2 𝑥 + 3
b.
𝑥+4
𝑥→−4 𝑥 2 + 2𝑥 − 8
lim
c.
𝑥+4
𝑥→2 𝑥 2 + 2𝑥 − 8
lim
Ex. 3: Evaluate the following limits.
a.
𝑥+3
𝑥→∞ 𝑥 2 − 2𝑥 + 1
lim
b.
2𝑥 + 3
𝑥→−∞ 3𝑥 + 1
lim
c.
lim 𝑒 −𝑥 + 2
d.
𝑥→∞
lim 𝑒 −𝑥 + 2
𝑥→−∞
Ex. 4: Evaluate the following limits.
a.
−4 sin 𝑥
𝑥→0
𝑥
lim
b.
cos 𝑥 − 1
+3
𝑥→−∞
𝑥
lim
c.
cos 𝑥
𝑥→0 sin 𝑥
lim
Learning Target 2-LI: Perform functional analysis, including limits.
2-LI
Concept: How to perform functional analysis.
𝐴(𝑥 ) =
𝑥 2 −7𝑥−18
𝑥 2 +5𝑥+6
Ex. 1: Analyze the following function.
𝑓(𝑥) =
3𝑥 2 −27
𝑥+3
Domain:
Range:
Asymptote(s):
Point(s) of Discontinuity:
x-intercept(s):
y-intercept(s):
End Behavior:
Ex. 2: Analyze the following function.
𝑓(𝑥) = 𝑒 𝑥−1 + 2
Domain:
Range:
Asymptote(s):
Point(s) of Discontinuity:
x-intercept(s):
y-intercept(s):
End Behavior:
Ex. 3: Analyze the following function.
𝑓(𝑥) = log 2 (𝑥 − 3)
Domain:
Range:
Asymptote(s):
Point(s) of Discontinuity:
x-intercept(s):
y-intercept(s):
End Behavior:
Ex. 4: Analyze the following function.
𝑓(𝑥) = 2√𝑥 − 1 + 4
Domain:
Range:
Asymptote(s):
Point(s) of Discontinuity:
x-intercept(s):
y-intercept(s):
End Behavior:
1-SQ
Learning Target 1-SQ: Generate a general term for, and find terms in arithmetic
and geometric sequences.
Concept: Arithmetic and Geometric Sequences.
Ex. 1: Find the next 3 terms in the sequence and the equation for the general term.
16 , 9 , 2 , −5 , . . .
Ex. 2: Find the next 3 terms in the sequence and the equation for the general term.
At the beginning
At one minute
At three minutes
At two minutes
At four minutes
Ex. 3: Find the 1st term and the general term for a geometric sequence where the eighth term
1
1
is − and the ninth term is .
3
9
Ex. 4: The 53rd term of an arithmetic sequence is -843 and the 54th term is -842.5. Find the
first terms and the general term.
Ex. 5: What is the 129th term for the following sequence: -8 , 12 , -18 , 27 , . . .
2-SQ
Learning Target 2-SQ: Find the sum of finite arithmetic, finite geometric, and
infinite geometric series.
Concept: Sum of a sequence is called a series.
Finite Series:
Infinite Series:
Ex. 1: Find the sum of the first 60 terms of a sequence where the first term is 9 and the
common difference is 5.
Ex. 2: Find the sum of the first ten terms of 16, −48, 144, −432, . . .
Ex. 3: Sum the first 32 terms of an arithmetic sequence where 𝑎1 = −12 and 𝑎32 = 174 .
Ex. 4: Find the sum of the first 8 terms of 14, −70, 350, −1750, . . .
Ex. 5: Sum the first 163 terms where 𝑑 = −
2
3
and 𝑎163 = 205 .
Ex. 6: Find the sum if it exists. If it does not exist, explain why.
4
36 + 12 + 4 + + ⋯
3
Ex. 7: Find the sum if it exists. If it does not exist, explain why.
16 + 8 + 0 + −8 + ⋯
Ex. 8: Find the sum if it exists. If it does not exist, explain why.
18 + 27 + 40.5 + ⋯
Ex. 9: Find the sum if it exists. If it does not exist, explain why.
100 + −80 + 64 + −51.2 + ⋯
Ex. 10: Use an infinite sum formula to write the repeating decimal as a fraction.
0. 2̅
Ex. 11: Use an infinite sum formula to write the repeating decimal as a fraction.
̅̅̅̅
0. 87
Ex. 12: Use an infinite sum formula to write the repeating decimal as a fraction.
̅̅̅̅
0.215
Ex. 13: Use an infinite sum formula to write the repeating decimal as a fraction.
5.16 + 0.00516 + 0.00000516 . . .
Learning Target 3-SQ: Generate a general term for a series.
3-SQ
Concept: LOOK FOR PATTERNS!!!
Ex. 1: Find the general term for each series.
a.
5
1
+
5
1∙2
+
5
1∙2∙3
+...
3
3
1
5
7
3
b. 1 + + + . . .
Ex. 2: Find the general term for each sequence.
a. 16 + 49 + 100 + 169 + . . .
b. 5 + 6 + 8 + 12 + 20 + . . .
Ex. 3: Find the general term for each series.
1
1
1
1
2
4
8
16
a. − + − +
−...
b.
1
10
1
3
5
10
− +
2
− + ...
5
Learning Target 4-SQ: Use sigma notation to formulate and evaluate series.
4-SQ
Concept: You can represent a series in compact form with Sigma Notation!
Ex. 1: Evaluate the series.
4
∑ 𝑛2 − 3
𝑛=1
Ex. 2: Evaluate the series.
7
∑ 3𝑘 + 1
𝑘=4
Ex. 3: Evaluate the series.
∞
2 𝑛−1
∑ 5 (− )
7
𝑛=1
Ex. 4: Evaluate the series.
∞
1 𝑛−1
∑2( )
3
𝑛=1
Ex. 5: Express in Sigma Notation.
15 + 24 + 35 + 48+ . . . +143
Ex. 6: Express in Sigma Notation.
16 + 19 + 22 + 25+ . . . +61
Ex. 7: Express in Sigma Notation.
2187 + 1458 + 972+ . . . +128
Ex. 8: Express in Sigma Notation.
6 + 24 + 120+ . . . + 40320