[email protected] MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical &

[email protected] • MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§10.1Inf

Series



Review §

Any QUESTIONS About• §9.4 More Differential Equation

Applications

Any QUESTIONS About HomeWork• §9.4 → HW-16

9.4



§10.1 Learning Goals

Determine convergence or divergence of an infinite series

Examine and use geometric series



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



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)



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



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, . . .



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



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



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



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



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



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



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



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



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



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



Sum & Multiple Rules



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



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.




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




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



WhiteBoard Work

Problems From §10.1• P49 Follow the

Bouncing Ball

http://misterknuckles.com/wp-content/uploads/2012/05/bouncingball.jpg

http://okc.net/wp-content/uploads/2014/03/bounce.jpg



All Done for Today

Series:ArithmeticGeometric



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22

a2 b2





Bouncing Ball

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&docid=zDqjDGnmS-tMOM&tbnid=ss7Mdy09v6XtIM:&ved=0CAUQjRw&url=http://www.iministries.org/blogarchive.aspx?blog_id=90645&site_id=10267&tag=Church%20Technology&ei=uNdKU_abEOz_yQHZ14EQ&bvm=bv.64542518,d.b2U&psig=AFQjCNExMnWKVx-QRdk8me5dzwW2lEIz_g&ust=1397500149702136















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[email protected] MTH16_Lec-14_sec_10-1_Infinite_Series.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical &