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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§10.1Inf
Series
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Bruce Mayer, PE Chabot College Mathematics
Review §
Any QUESTIONS About• §9.4 More Differential Equation
Applications
Any QUESTIONS About HomeWork• §9.4 → HW-16
9.4
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Bruce Mayer, PE Chabot College Mathematics
§10.1 Learning Goals
Determine convergence or divergence of an infinite series
Examine and use geometric series
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Bruce Mayer, PE Chabot College Mathematics
Infinite SEQUENCE
An infinite sequence is a function which • Has the domain of all NATURAL Numbers• A constant Mathematical Relationship
between adjacent elements
a1, a2, a3, a4, . . . , an, . . .
Elements
The 1st 3 elements of the sequence an = 2n2
a1 = 2(1)2 = 2
a2 = 2(2)2 = 8
a3 = 2(3)2 = 18
FiniteSequence
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Bruce Mayer, PE Chabot College Mathematics
Arithmetic vs. Geometric Seq
ARITHMETIC Sequence → Repeatedly ADD a number, d (a difference), to some initial value, a
GEOMETRIC Sequence → start with a number a and repeatedly MULTIPLY by a fixed nonzero constant value, r ( a ratio)
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Bruce Mayer, PE Chabot College Mathematics
GeoMetric Sequence
A sequence is GEOMETRIC if the ratios of consecutive terms are the same.
2, 8, 32, 128, 512, . . . Geometric Sequence
The common ratio, r, is 4
82
4
328
4
12832
4
512128
4
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Bruce Mayer, PE Chabot College Mathematics
GeoMetric Sequence: “nth” Term The nth term of a geometric sequence
has the form: an = a1rn−1
• where r is the common ratio of consecutive terms of the sequence
Example
• The nth term is 15(5n-1)
a1 = 15
75 515
r
a2 = 15(5)
a3 = 15(52)
a4 = 15(53)
15, 75, 375, 1875, . . .
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Bruce Mayer, PE Chabot College Mathematics
Example GeoMetric Seq
Determine a1, r ,and the nth term for the GeoMetric Sequence
Recognize: a1, = 1, and r = ⅓
The nth term is: an = (⅓)n–1
1 1 1 11, , , , , . . .
3 9 27 81
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Bruce Mayer, PE Chabot College Mathematics
Summation Notation
Represent the first n terms of a sequence by summation notation.
Example
1 2 3 41
n
i ni
a a a a a a
index of summation
upper limit of summation
lower limit of summation
5
1
4n
n
1 2 3 4 54 4 4 4 4 4 16 64 256 1024 1364
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Bruce Mayer, PE Chabot College Mathematics
Finite Sum for GeoMetric Sequence
The Sum of a Finite Geometric Sequence Given By 1
1 11
1 .1
n nin
i
rS a r ar
5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ?
n = 8
a1 = 5 5210r
1
81 11
221
5n
nrS ar
1 25651 2 2555
1 1275
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Bruce Mayer, PE Chabot College Mathematics
INFinite Sum for GeoMetric Seq
The sum of the terms of an INfinite geometric sequence is called a Geometric Series
If |r| < 1, then the infinite geometric series
has the Sum:
If |r| ≥ 1, then the infinite geometric series Does NOT have a Sum (it Diverges)
a1 + a1r + a1r2 + a1r3 + . . . + a1rn-1 + . . .
11
0
.1
i
i
aS a r
r
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Bruce Mayer, PE Chabot College Mathematics
Example Infinite Series
Find the sum of Recognize:
Thus the Series Sum:
1 13 13 9
13
r
1
1a
Sr
3
1 13
3 31 413 3
3 934 4
9 .4
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Bruce Mayer, PE Chabot College Mathematics
nth Partial Sum of a Series
The General form of an Infinite Series
Then a Finite Fragment of the Sum is called the nth Partial Sum →
• Where n is simply any Natural Number (say 537)
1 2 31
... ...n ii
S a a a a a
1 2 31
...n
n n ii
S a a a a a
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Bruce Mayer, PE Chabot College Mathematics
Convergence vs. Divergence
An Infinite series with nth Partial Sum
CONVERGES to sum S if S is a Finite Number such that
In this Case
The Series is said to DIVERGE when• i.e., the Limit Does Not Exist
1 2 31
...n
n n ii
S a a a a a
SSnn
lim
1i
i Sa
n
nSlim
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Bruce Mayer, PE Chabot College Mathematics
Example Divergence
This Series DIVERGES
Note that the quantity {1+3n} increases without bound
Then the partial Sum:
Always Increase as K increases
K
n
n1
31
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Bruce Mayer, PE Chabot College Mathematics
Example Convergence
It is known that the following Leibniz series converges to the value π/4 as n→∞ for the Partial Sum:
This Convergence is difficult to Prove, so Check numerically for n: 1→200
0 20 40 60 80 1000.65
0.7
0.75
0.8
0.85
0.9
n
Su
m &
/4
MTH16 • Leibniz Series
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Bruce Mayer, PE Chabot College Mathematics
MATLAB Code for Leibniz% Bruce Mayer, PE% ENGR25 * 12Apr14% file = MTE_Leibinz_Series_1404.m%clear; clc; clf;%N = 100 % the Number terms in the Sum N+1for n = 1:N k = 0:n; S(n) = sum((-1).^k.*(1./(2*k+1)));end% Calc DIFFERENCE compare to pi/4 %% The y = PI Lineszxh = [0 N]; zyh = [pi/4 pi/4];%whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenaxes; set(gca,'FontSize',12);plot((1:N),S,'b', 'LineWidth',1.5), grid,... xlabel('\fontsize{14}n'), ylabel('\fontsize{14}Sum & \pi/4'),... title(['\fontsize{16}MTH16 • Leibniz Series',])hold onplot(zxh,zyh, 'g', 'LineWidth', 2)hold off
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Bruce Mayer, PE Chabot College Mathematics
Sum & Multiple Rules
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Bruce Mayer, PE Chabot College Mathematics
Example Use Sum & Mult Rules
Assume that this Series Converges to 4:
Use this information to find the value of
SOLUTION Using properties
of convergent infinite series, find →
1
2
3
nn na
24
2
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Bruce Mayer, PE Chabot College Mathematics
Example Negative Advertising
A Social Science study suggests that Negative political ads work, but only over short periods of time. Assume that a Negative ad influences the vote of 500 voters, but that influence decays at an instantaneous rate of 40% per day.
Find the number of influenced voters (a) as a partial sum if Negative ads are run each day for a week and (b) if the ads were continued at a daily rate indefinitely.
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Bruce Mayer, PE Chabot College Mathematics
Example Negative Advertising
SOLUTION:
a) Each ad influences 500 voters initially, and then drops off precipitously: only a fraction of e−0.40t total voters remain influenced after t days. Thus the partial sum over a week of advertising:
• Thus The ads influence about 955 voters during the week.
8.954
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Bruce Mayer, PE Chabot College Mathematics
Example Negative Advertising
b) The infinite sum calculates the effect of continuing the ads indefinitely
So The ads influence about 1017 voters if continued indefinitely - less than 100 additional votes compared to running the ads for only one week.
6.1016
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Bruce Mayer, PE Chabot College Mathematics
WhiteBoard Work
Problems From §10.1• P49 Follow the
Bouncing Ball
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Bruce Mayer, PE Chabot College Mathematics
All Done for Today
Series:ArithmeticGeometric
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
Appendix
–
srsrsr 22
a2 b2
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
Bouncing Ball
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics