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
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
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.
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
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
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
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
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
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.
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
Sequences
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
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
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
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
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
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
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
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
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
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
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
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
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
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
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
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, …
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
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.
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
Convergent and Divergent Sequences
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
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
Convergent and Divergent Sequences
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
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
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
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
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
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
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
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
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
Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
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
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