Infinite sequence & series 1st lecture

Copyright © 2013 CHEP, P. U. Lahore. Lecture-1

Infinite Series

Tutor: Dr. Tariq MahmoodAssistant ProfessorCentre for High Energy

PhysicsUniversity of the Punjab

Class: B.Sc (Hons.)Computational Physics


Tutor’s Brief IntroductionName: Dr. Tariq Mahmood KhanDesignation: Assistant ProfessorQualification: Ph.D (BIT, Beijing, P. R. China) in

Computational Materials PhysicsM.Phil (CHEP, P. U., Lahore, Pakistan)

Email: [email protected] Publications: More than 30 articles have been published in International SCI

journals with good impact factors (Physica B: Condensed Matter, Materials Letter, The Journal of Physical Chemistry A, Electrochimica Acta, Materials Research Bulletin, Journal of Alloys and Compounds, Journal of Nanoscience and Nanotechnology, Solid State Sciences, Materials Chemistry and Physics , Materials Research Bulletin , Current NanoScience, Thin Solid Films, Sains Malaysiana, Journal of Optoelectronics and Advance Materials.

mailto:[email protected]


Course DescriptionClasses: 34-36 (2 credit hours)Total Marks: 100Assignments: 25Mid Term: 35Final Term: 40

Note: Students with less than 75% attendance will not able to sit in the exam.

Book: Calculus, Ninth Edition By Thomas and FinneyThomas’ Calculus 11th Edition By Maurice D. Weir

et al

Chapter: Chapter 8 and 11


SyllabusLimits of Sequences of NumbersTheorems for Calculating Limits of SequencesInfinite SeriesThe Integral Test for Series of Nonnegative TermsComparison Tests for series of Nonnegative TermsThe Ratio and Root Tests for Series of Nonnegative TermsAlternating Series, Absolute and Conditional ConvergencePower SeriesTaylor and Maclaurin SeriesConvergence of Taylor Series; Error EstimatesApplication of Power Series


Applications

Infinite sequences and series are important in physics and engineering. One of the most well-known is the Fourier series , which can mathematically define certain signal waveforms. In Materials Physics, infinite series are used to calculate different calculations (Electric, mechanical, optical, etc) in the form of wave functions.

http://whatis.techtarget.com/definition/Fourier-series

http://searchcio-midmarket.techtarget.com/definition/waveform


Sequences


Objectives List the terms of a sequence.

Determine whether a sequence converges or diverges.

Write a formula for the nth term of a sequence.

Recursion formula


Sequences

A sequence is defined as a function whose domain is the set of positive integers. Although a sequence is a function, it is common to represent sequences by subscript notation rather than by the standard function notation. For instance, in the sequence

1 is mapped onto a1, 2 is mapped onto a2, and so on. The numbers a1, a2, a3, . . ., an, . . . are the terms of the sequence. The number an is the nth term of the sequence, and the entire sequence is denoted by {an}.

Sequence


Sequences

f (x) notation is not used for sequences.

Write Sequences are written as ordered lists

a1 is the first element, a2 the second element, and so on

A sequence is a function that has a set of natural numbers as its domain.

( )na f n

1 2 3, , , ...a a a


Example 1 – Listing the Terms of a Sequence

a. The terms of the sequence {an} = {3 + (–1)n} are 3 + (–1)1, 3 + (–1)2, 3 + (–1)3, 3 + (–1)4, . . .

2, 4, 2, 4, . . . .

b. The terms of the sequence {bn} are


Example 1 – Listing the Terms of a Sequence

c. The terms of the sequence {cn} are

d. The terms of the recursively defined sequence {dn}, where d1 = 25 and dn + 1 = dn – 5, are

25, 25 – 5 = 20, 20 – 5 = 15, 15 – 5 = 10,. . . . .

cont’d


Sequences

A sequence is often specified by giving a formula for

the general term or nth term, an.

Example Find the first four terms for the sequence

Solution

1

2n

na

n

1 2(1 1) /(1 2) 2 / 3, (2 1) /(2 2) 3/ 4a a

3 4(3 1) /(3 2) 4 / 5, (4 1) /(4 2) 5 / 6a a


Graphing Sequences

The graph of a sequence, an, is the graph of the

discrete points (n, an) for n = 1, 2, 3, …

Example Graph the sequence an = 2n.

Solution


Sequences

A finite sequence has domain the finite set {1, 2, 3, …, n} for some natural number n.Example 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

An infinite sequence has domain {1, 2, 3, …}, the set of all natural

numbers.Example 1, 2, 4, 8, 16, 32, …


Convergent and Divergent Sequences

A convergent sequence is one whose terms get closer and closer to a some real number. The sequence is said to converge to that number.

A sequence that is not convergent is said to be divergent.



Example The sequence converges to 0.

The terms of the sequence 1, 0.5, 0.33.., 0.25, …

grow smaller and smaller approaching 0. This can be

seen graphically.

1na n



Example The sequence is divergent.

The terms grow large without bound

1, 4, 9, 16, 25, 36, 49, 64, …

and do not approach any one number.

2na n


Sequences and Recursion Formulas

A recursion formula or recursive definition defines a sequence bySpecifying the first few terms of the sequence

Using a formula to specify subsequent terms in terms of preceding terms.

OR term can be calculated directly from the value of

n. But sequences are defined recursively by giving The value(s) of the initial term(s) The rule, called a recursion formula, for calculating any

later term from terms that precede it.

na


Using a Recursion Formula

Example Find the first four terms of the sequence a1 = 4; for n>1, an = 2an-1 + 1

Solution We know a1 = 4.

Since an = 2an-1 + 1

2 1

3 2

4 3

2 1 2 4 1 9

2 1 2 9 1 19

2 1 2 19 1 39

a a

a a

a a


Applications of Sequences

Example The winter moth population in thousands

per acre in year n, is modeled by

for n > 2

(a) Give a table of values for n = 1, 2, 3, …, 10

(b) Graph the sequence.

21 1 11, 2.85 .19n n na a a a


Applications of Sequences

Solution(a)

(b)Note the population stabilizes near a value of 9.7 thousand insects per acre.

n 1 2 3 4 5 6

an 1 2.66 6.24 10.4 9.11 10.2

n 7 8 9 10

an 9.31 10.1 9.43 9.98


Assignment-1Let an and bn be sequences of real numbers and

let A and B be real numbers. The following rules hold ifand Sum Rule: Difference Rule: Product Rule: Constant Multiple Rule:

(any number of k)

Quotient Rule: if

lim n nt a A

lim n nt b B lim ( )n n nt a b A B

lim ( )n n nt a b A B lim ( . ) .n n nt a b A B

lim ( . ) .n nt k b k B

lim nn

n

a At

b B 0B

Education

Infinite sequence & series 1st lecture